The document discusses limits and derivatives. It explains that in calculating the derivative of f(x)=x^2 - 2x + 2, the slope formula was simplified. As h approaches 0, the chords slide towards the tangent line, so the slope at (x,f(x)) is 2x-2. It then provides definitions and explanations for what it means for a variable to approach 0 from the right, left, or in general, to clarify the procedure of obtaining slopes using limits.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
Get an inside look at the latest Neo4j innovations that enable relationship-driven intelligence at scale. Learn more about the newest cloud integrations and product enhancements that make Neo4j an essential choice for developers building apps with interconnected data and generative AI.
GridMate - End to end testing is a critical piece to ensure quality and avoid...ThomasParaiso2
End to end testing is a critical piece to ensure quality and avoid regressions. In this session, we share our journey building an E2E testing pipeline for GridMate components (LWC and Aura) using Cypress, JSForce, FakerJS…
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
2. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)).
y = x2–2x+2
(x, f(x))
x
3. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)).
y = x2–2x+2
(x+h, f(x+h)
(x, f(x))
f(x+h)–f(x)
h
x x + h
4. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)).
y = x2–2x+2
(x+h, f(x+h)
(x, f(x))
f(x+h)–f(x)
h
x x + h
5. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)). We obtained the
cord–slope–formula 2x – 2 + h.
y = x2–2x+2
(x+h, f(x+h)
slope = 2x–2+h
(x, f(x))
f(x+h)–f(x)
h
x x + h
6. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)). We obtained the
cord–slope–formula 2x – 2 + h.
We reason that as the values of
h shrinks to 0,
y = x2–2x+2
(x+h, f(x+h)
slope = 2x–2+h
(x, f(x))
f(x+h)–f(x)
h
x x + h
7. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)). We obtained the
cord–slope–formula 2x – 2 + h.
We reason that as the values of
h shrinks to 0, the cords slide
towards the tangent line
y = x2–2x+2
(x+h, f(x+h)
slope = 2x–2+h
(x, f(x))
f(x+h)–f(x)
h
x x + h
8. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)). We obtained the
cord–slope–formula 2x – 2 + h.
We reason that as the values of
h shrinks to 0, the cords slide
towards the tangent line so the
slope at (x, f(x)) must be 2x – 2
because h “fades” to 0.
y = x2–2x+2
(x+h, f(x+h)
slope = 2x–2+h
(x, f(x))
f(x+h)–f(x)
h
x x + h
9. Limits I
In the calculation of the
derivative of f(x) = x2 – 2x + 2,
we simplified the slope
(difference-quotient) formula of
the cords with one end fixed at
(x, f(x)) and the other end at
(x+h, f(x+h)). We obtained the
cord–slope–formula 2x – 2 + h.
We reason that as the values of
h shrinks to 0, the cords slide
towards the tangent line so the
slope at (x, f(x)) must be 2x – 2
because h “fades” to 0.
y = x2–2x+2
(x+h, f(x+h)
slope = 2x–2+h
(x, f(x))
f(x+h)–f(x)
h
x x + h
We use the language of “limits” to
clarify this procedure of obtaining slopes .
10. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
11. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
12. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
13. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
0 x’s
14. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
for any ϵ > 0
0 ϵ x’s
15. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
for any ϵ > 0 only finitely x’s are outside
0 ϵ x’s
16. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
for any ϵ > 0 only finitely x’s are outside
0 ϵ x’s
The point here is that no matter
how small the interval (0, ϵ) is,
most of the x’s are in (0, ϵ).
17. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
for any ϵ > 0 only finitely x’s are outside
0 ϵ x’s
We say “as x goes to 0+ we get that …” we mean that
for “every sequence {xi} where xi 0+ we would obtain
the result mentioned”.
18. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
for any ϵ > 0 only finitely x’s are outside
0 ϵ x’s
We say “as x goes to 0+ we get that …” we mean that
for “every sequence {xi} where xi 0+ we would obtain
the result mentioned”. So “as x 0+, x + 2 2” means
that for any sequence xi 0+ we get xi + 2 2.
19. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (0, ϵ)”.
for any ϵ > 0 only finitely x’s are outside
We say “as x goes to 0+ we get that …” we mean that
for “every sequence {xi} where xi 0+ we would obtain
the result mentioned”. So “as x 0+, x + 2 2” means
that for any sequence xi 0+ we get xi + 2 2.
We write this as lim (x + 2) = 2 or lim (x + 2) = 2.
0+
0 ϵ x’s
x 0+
21. Limits I
Similarly we define
“x approaches 0 from the – (left) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
22. Limits I
Similarly we define
“x approaches 0 from the – (left) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ,0)”.
only finitely x’s are outside for any ϵ > 0
x’s –ϵ 0
23. Limits I
Similarly we define
“x approaches 0 from the – (left) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ,0)”.
only finitely x’s are outside for any ϵ > 0
x’s –ϵ 0
We say “as x goes to 0– we get that …” we mean that
for “every sequence {xi} where xi 0– we would
obtain the result mentioned”.
24. Limits I
Similarly we define
“x approaches 0 from the – (left) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ,0)”.
only finitely x’s are outside for any ϵ > 0
x’s –ϵ 0
We say “as x goes to 0– we get that …” we mean that
for “every sequence {xi} where xi 0– we would
obtain the result mentioned”. So “as x 0–, x + 2 2”
means that for any sequence xi 0– we get xi + 2 2.
25. Similarly we define
“x approaches 0 from the – (left) side”.
We say the sequence {xi} = {x1, x2, x3, .. }
“goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ,0)”.
only finitely x’s are outside for any ϵ > 0
We say “as x goes to 0– we get that …” we mean that
for “every sequence {xi} where xi 0– we would
obtain the result mentioned”. So “as x 0–, x + 2 2”
means that for any sequence xi 0– we get xi + 2 2.
We write this as lim (x + 2) = 2 or lim (x + 2) = 2.
0–
Limits I
x 0–
x’s –ϵ 0
26. Limits I
Finally we say that
“xi goes to 0” or “xi 0” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ, ϵ)”.
27. Limits I
Finally we say that
“xi goes to 0” or “xi 0” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ, ϵ)”.
x’s –ϵ 0
ϵ x’s
only finitely many x’s are outside
28. Limits I
Finally we say that
“xi goes to 0” or “xi 0” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ, ϵ)”.
x’s –ϵ 0
ϵ x’s
only finitely many x’s are outside
We say “as x goes to 0 we get that …” we mean that
for “every sequence {xi} where xi 0 we obtain the
result mentioned”.
29. Finally we say that
“xi goes to 0” or “xi 0” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ, ϵ)”.
x’s –ϵ 0
ϵ x’s
only finitely many x’s are outside
We say “as x goes to 0 we get that …” we mean that
for “every sequence {xi} where xi 0 we obtain the
result mentioned”.
Hence lim x + 1 = 1/–1 = –1.
0 2x – 1
Limits I
30. Limits I
Finally we say that
“xi goes to 0” or “xi 0” where i = 1, 2, 3…
“if for any number ϵ > 0 only finitely many of the x’s
are outside of the neighborhood (–ϵ, ϵ)”.
x’s –ϵ 0
ϵ x’s
only finitely many x’s are outside
We say “as x goes to 0 we get that …” we mean that
for “every sequence {xi} where xi 0 we obtain the
result mentioned”.
Hence lim x + 1 = 1/–1 = –1.
0 2x – 1
Let’s generalize this to “x a” where a is any number.
31. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
32. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
a a+ϵ x’s
33. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
a a+ϵ x’s
The notation “xi a–” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a )."
34. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
a a+ϵ x’s
The notation “xi a–” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a )."
x’s a–ϵ a
35. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
a a+ϵ x’s
The notation “xi a–” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a )."
x’s a–ϵ a
The notation “xi a” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a + ϵ ).”
36. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
a a+ϵ x’s
The notation “xi a–” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a )."
x’s a–ϵ a
The notation “xi a” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a + ϵ ).”
x’s a–ϵ a a+ϵ x’s
37. Limits I
The notation “xi a+” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a, a + ϵ )."
a a+ϵ x’s
The notation “xi a–” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a )."
x’s a–ϵ a
The notation “xi a” where {xi} is a sequence of
numbers means that “for every ϵ > 0, only finitely
many x’s are outside of the interval (a – ϵ, a + ϵ ).”
x’s a–ϵ a a+ϵ x’s
We say lim f(x) = L if f(xi) L for every xi a (or a±).
a (or a±)
38. Limits I
The following statements of limits as x a are true.
39. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
Limits I
40. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
Limits I
a
41. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
* lim x = a
Limits I
a
a
42. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
* lim x = a
(e.g. lim x = 5)
Limits I
a
a
5
43. Limits I
The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
a
* lim x = a
a
(e.g. lim x = 5)
* lim cx = ca where c is any number.
a
5
44. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
* lim x = a
(e.g. lim x = 5)
* lim cx = ca where c is any number.
(e.g. lim 3x = 15)
Limits I
a
a
a
5
5
45. Limits I
The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
a
* lim x = a
a
(e.g. lim x = 5)
* lim cx = ca where c is any number.
a
(e.g. lim 3x = 15)
* lim (xp) = (lim x)p = ap provided ap is well defined.
a
5
5
a
46. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
* lim x = a
(e.g. lim x = 5)
* lim cx = ca where c is any number.
(e.g. lim 3x = 15)
* lim (xp) = (lim x)p = ap provided ap is well defined.
(e.g. lim x½ = 5)
Limits I
a
a
a
a
5
5
a
25
47. Limits I
The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
(e.g lim 5 = 5)
a
* lim x = a
a
(e.g. lim x = 5)
5
* lim cx = ca where c is any number.
a
(e.g. lim 3x = 15)
5
* lim (xp) = (lim x)p = ap provided ap is well defined.
a
a
(e.g. lim x½ = 5)
25
Reminder: the same statements hold true for x a±.
48. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
49. Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
Limits I
50. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0.
a
51. Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
Limits I
a
52. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
a
In fact, if f(x) is an elementary function and f(a) is
well defined, i.e. a is in the domain of f(x),
53. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
a
In fact, if f(x) is an elementary function and f(a) is
well defined, i.e. a is in the domain of f(x), then
lim f(x) = f(a) as x a or x a±,
54. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
a
In fact, if f(x) is an elementary function and f(a) is
well defined, i.e. a is in the domain of f(x), then
lim f(x) = f(a) as x a or x a±,
provided the selections of such x’s are possible.
55. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
a
In fact, if f(x) is an elementary function and f(a) is
well defined, i.e. a is in the domain of f(x), then
lim f(x) = f(a) as x a or x a±,
provided the selections of such x’s are possible.
For example, the domain of the function f(x) = √x is
0 <– x.
56. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
a
In fact, if f(x) is an elementary function and f(a) is
well defined, i.e. a is in the domain of f(x), then
lim f(x) = f(a) as x a or x a±,
provided the selections of such x’s are possible.
For example, the domain of the function f(x) = √x is
0 <– x. Hence lim√x = √a for 0 < a. a
57. Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.
1. lim P(x) = P(a)
a
2. lim = P(x)
Q(x)
P(a)
Q(a)
, Q(a) = 0. (e.g. lim x + 2
x – 3 1
= –3/2)
In fact, if f(x) is an elementary function and f(a) is
well defined, i.e. a is in the domain of f(x), then
lim f(x) = f(a) as x a or x a±,
provided the selections of such x’s are possible.
For example, the domain of the function f(x) = √x is
0 <– x. Hence lim√x = √a for 0 < a. a
However at a = 0, we could only
have lim √x = 0 = f(0) as shown.
y = x1/2
0+
(but not 0)
Limits I
a
59. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”.
60. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”. The domain of
the 1/x is the set of all numbers x except x = 0.
61. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”. The domain of
the 1/x is the set of all numbers x except x = 0.
Although we can’t evaluate 1/x with x = 0, we still
know the behavior of f(x) as x takes on small values
that are close to 0 as demonstrated in the table
below.
62. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”. The domain of
the 1/x is the set of all numbers x except x = 0.
Although we can’t evaluate 1/x with x = 0, we still
know the behavior of f(x) as x takes on small values
that are close to 0 as demonstrated in the table
below.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 ?
63. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”. The domain of
the 1/x is the set of all numbers x except x = 0.
Although we can’t evaluate 1/x with x = 0, we still
know the behavior of f(x) as x takes on small values
that are close to 0 as demonstrated in the table
below.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 ?
From the table we see that the corresponding 1/x
expands unboundedly to ∞.
64. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”. The domain of
the 1/x is the set of all numbers x except x = 0.
Although we can’t evaluate 1/x with x = 0, we still
know the behavior of f(x) as x takes on small values
that are close to 0 as demonstrated in the table
below.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 ?
From the table we see that the corresponding 1/x
expands unboundedly to ∞. Let’s make “expands
unboundedly to ∞” more precise.
65. Limits I
A set of infinitely many numbers S = {x’s} is said to be
bounded above if there is a number R such that x < R
for all the numbers x in the set S.
66. Limits I
A set of infinitely many numbers S = {x’s} is said to be
bounded above if there is a number R such that x < R
for all the numbers x in the set S. The “R” stands for
“to the right” as shown.
x’s R
67. Limits I
A set of infinitely many numbers S = {x’s} is said to be
bounded above if there is a number R such that x < R
for all the numbers x in the set S. The “R” stands for
“to the right” as shown.
x’s R
A set of numbers S = {x} is said to be
bounded below if there is a number L such that L < x
for all the x in the set.
68. Limits I
A set of infinitely many numbers S = {x’s} is said to be
bounded above if there is a number R such that x < R
for all the numbers x in the set S. The “R” stands for
“to the right” as shown.
x’s R
A set of numbers S = {x} is said to be
bounded below if there is a number L such that L < x
for all the x in the set. The “L” stands for “to the left”
as shown.
L x’s
69. Limits I
A set of infinitely many numbers S = {x’s} is said to be
bounded above if there is a number R such that x < R
for all the numbers x in the set S. The “R” stands for
“to the right” as shown.
x’s R
A set of numbers S = {x} is said to be
bounded below if there is a number L such that L < x
for all the x in the set. The “L” stands for “to the left”
as shown.
L x’s
We say that the interval (L, R) is bounded above
and below, or that it is bounded.
L x’s R
70. Limits I
The 1/x–values on the list is bounded below – a
lower bound L = 0 < 1/x.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 …
71. Limits I
The 1/x–values on the list is bounded below – a
lower bound L = 0 < 1/x. However the list is not
bounded above.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 …
72. Limits I
The 1/x–values on the list is bounded below – a
lower bound L = 0 < 1/x. However the list is not
bounded above.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 …
This list has the following property.
73. Limits I
The 1/x–values on the list is bounded below – a
lower bound L = 0 < 1/x. However the list is not
bounded above.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 …
This list has the following property.
For any large number G we select, there are only
finitely many entries that are smaller than G.
74. Limits I
The 1/x–values on the list is bounded below – a
lower bound L = 0 < 1/x. However the list is not
bounded above.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 …
This list has the following property.
For any large number G we select, there are only
finitely many entries that are smaller than G.
For example, if G = 10100 then only entries to the left
of the 100th entry are less than G.
75. Limits I
The 1/x–values on the list is bounded below – a
lower bound L = 0 < 1/x. However the list is not
bounded above.
x 0.1 0.01 0.001 0.0001 0+
f(x) = 1/x 10 100 1,000 10,000 …
This list has the following property.
For any large number G we select, there are only
finitely many entries that are smaller than G.
For example, if G = 10100 then only entries to the left
of the 100th entry are less than G.
x 0.1 0.01 0.001 0.0001 … 100th entry …
f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries
only these entries are < 10100
76. Limits I
x 0.1 0.01 0.001 0.0001 … 100th entry …
f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries
In the language of limits, we say that
lim 1/x = ∞
0+
77. Limits I
x 0.1 0.01 0.001 0.0001 … 100th entry …
f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries
In the language of limits, we say that
lim 1/x = ∞
0+
and it is read as “the limit of 1/x, as x goes to 0+ is ∞”.
78. Limits I
x 0.1 0.01 0.001 0.0001 … 100th entry …
f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries
In the language of limits, we say that
lim 1/x = ∞
0+
and it is read as “the limit of 1/x, as x goes to 0+ is ∞”.
In a similar fashion we have that
“the limit of 1/x, as x goes to 0– is –∞” as
lim 1/x = –∞
0–
79. x 0.1 0.01 0.001 0.0001 … 100th entry …
f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries
In the language of limits, we say that
lim 1/x = ∞
0+
and it is read as “the limit of 1/x, as x goes to 0+ is ∞”.
In a similar fashion we have that
“the limit of 1/x, as x goes to 0– is –∞” as
lim 1/x = –∞
0–
However lim 1/x is undefined (UDF) because the
0
signs of 1/x is unknown so no general conclusion
may be made except that |1/x| ∞.
Limits I
80. x 0.1 0.01 0.001 0.0001 … 100th entry …
f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries
In the language of limits, we say that
lim 1/x = ∞
0+
and it is read as “the limit of 1/x, as x goes to 0+ is ∞”.
In a similar fashion we have that
“the limit of 1/x, as x goes to 0– is –∞” as
lim 1/x = –∞
0–
However lim 1/x is undefined (UDF) because the
0
signs of 1/x is unknown so no general conclusion
may be made except that |1/x| ∞. The behavior of
1/x may fluctuate wildly depending on the selections
of the x’s.
Limits I
81. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L,
82. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
83. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0.
∞
Hence
–∞
84. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x.
85. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
86. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
87. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
0+ As x 0+, lim 1/x = ∞
88. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
89. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
y = 1/x
x= 0: Vertical
Asymptote
90. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
y = 1/x
x= 0: Vertical
Asymptote
91. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
y = 1/x
x= 0: Vertical
Asymptote
92. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
y = 1/x
x= 0: Vertical
Asymptote
93. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
y = 1/x
x= 0: Vertical
Asymptote
94. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
II. The two “ends” of the line. y = 1/x
As x ∞, lim 1/x = 0+
∞
As x –∞, lim 1/x = 0–
–∞
x= 0: Vertical
Asymptote
95. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
II. The two “ends” of the line. y = 1/x
As x ∞, lim 1/x = 0+
∞
As x –∞, lim 1/x = 0–
–∞
x= 0: Vertical
Asymptote
96. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
II. The two “ends” of the line. y = 1/x
As x ∞, lim 1/x = 0+
∞
As x –∞, lim 1/x = 0–
–∞
x= 0: Vertical
Asymptote
97. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
II. The two “ends” of the line. y = 1/x
As x ∞, lim 1/x = 0+
∞
As x –∞, lim 1/x = 0–
–∞
x= 0: Vertical
Asymptote
98. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
II. The two “ends” of the line. y = 1/x
As x ∞, lim 1/x = 0+
∞
As x –∞, lim 1/x = 0–
–∞
x= 0: Vertical
Asymptote
99. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L, and we also
say that “as x –∞, f(x) L” if for every sequence
xi –∞ we have that f(xi) L.
lim 1/x = lim 1/x = 0. Let’s summarize the
∞
Hence
–∞
“boundary behaviors” of 1/x. The domain of 1/x is
x = 0 so its “boundary” consists of the following.
I. The vertical asymptote x = 0.
As x 0+, lim 1/x = ∞
0+
As x 0–, lim 1/x = –∞
0–
II. The two “ends” of the line. y = 1/x
As x ∞, lim 1/x = 0+
∞
As x –∞, lim 1/x = 0–
–∞
x= 0: Vertical
Asymptote
y = 0: Horizontal
Asymptote
101. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement.
102. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement. It represents the
behavior of a list of endless numbers, specifically
eventually “almost all” of the numbers are larger than
any imaginable numbers.
103. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement. It represents the
behavior of a list of endless numbers, specifically
eventually “almost all” of the numbers are larger than
any imaginable numbers. Hence we say that
“the sequence 1, 2, 3, .. goes to ∞” or that “the
sequence –1, –2, –3, .. goes to –∞”.
104. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement. It represents the
behavior of a list of endless numbers, specifically
eventually “almost all” of the numbers are larger than
any imaginable numbers. Hence we say that
“the sequence 1, 2, 3, .. goes to ∞” or that “the
sequence –1, –2, –3, .. goes to –∞”.
If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,..
the resulting sequence still goes to ∞.
105. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement. It represents the
behavior of a list of endless numbers, specifically
eventually “almost all” of the numbers are larger than
any imaginable numbers. Hence we say that
“the sequence 1, 2, 3, .. goes to ∞” or that “the
sequence –1, –2, –3, .. goes to –∞”.
If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,..
the resulting sequence still goes to ∞.
In fact, given any sequence of xi such that xi ∞,
then cxi ∞ for any 0 < c.
106. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement. It represents the
behavior of a list of endless numbers, specifically
eventually “almost all” of the numbers are larger than
any imaginable numbers. Hence we say that
“the sequence 1, 2, 3, .. goes to ∞” or that “the
sequence –1, –2, –3, .. goes to –∞”.
If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,..
the resulting sequence still goes to ∞.
In fact, given any sequence of xi such that xi ∞,
then cxi ∞ for any 0 < c. In short, we say that
c* ∞ = ∞ for any constant c > 0.
107. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement. It represents the
behavior of a list of endless numbers, specifically
eventually “almost all” of the numbers are larger than
any imaginable numbers. Hence we say that
“the sequence 1, 2, 3, .. goes to ∞” or that “the
sequence –1, –2, –3, .. goes to –∞”.
If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,..
the resulting sequence still goes to ∞.
In fact, given any sequence of xi such that xi ∞,
then cxi ∞ for any 0 < c. In short, we say that
c* ∞ = ∞ for any constant c > 0.
We summarize these facts about ∞ below.
109. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
2. ∞ * ∞ = ∞
3. c * ∞ = ∞ for any constant c > 0.
4. c / ∞ = 0 for any constant c.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
110. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞.
2. ∞ * ∞ = ∞
3. c * ∞ = ∞ for any constant c > 0.
4. c / ∞ = 0 for any constant c.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
111. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.)
2. ∞ * ∞ = ∞
3. c * ∞ = ∞ for any constant c > 0.
4. c / ∞ = 0 for any constant c.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
112. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.)
2. ∞ * ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
so lim (x * x2) = lim x * lim x2 = ∞.
3. c * ∞ = ∞ for any constant c > 0.
4. c / ∞ = 0 for any constant c.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
113. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.)
2. ∞ * ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
so lim (x * x2) = lim x * lim x2 = ∞. (Not true for “/“.)
3. c * ∞ = ∞ for any constant c > 0.
4. c / ∞ = 0 for any constant c.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
114. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.)
2. ∞ * ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
so lim (x * x2) = lim x * lim x2 = ∞.
3. c * ∞ = ∞ for any constant c > 0.
(Not true for “/“.)
As x goes to ∞, lim x = ∞, so lim 3x = ∞.
4. c / ∞ = 0 for any constant c.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
115. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.)
2. ∞ * ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
so lim (x * x2) = lim x * lim x2 = ∞.
3. c * ∞ = ∞ for any constant c > 0.
(Not true for “/“.)
As x goes to ∞, lim x = ∞, so lim 3x = ∞.
4. c / ∞ = 0 for any constant c.
As x goes to ∞, lim x = ∞, so lim 3/x = 0.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
116. Arithmetic of ∞
Limits I
1. ∞ + ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence
lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.)
2. ∞ * ∞ = ∞
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
so lim (x * x2) = lim x * lim x2 = ∞.
3. c * ∞ = ∞ for any constant c > 0.
(Not true for “/“.)
As x goes to ∞, lim x = ∞, so lim 3x = ∞.
4. c / ∞ = 0 for any constant c.
As x goes to ∞, lim x = ∞, so lim 3/x = 0.
5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
As x goes to ∞, lim 2x = ∞ and lim (½)x = 0.
117. Limits I
The following situations of limits are inconclusive.
118. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
119. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
120. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
121. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
122. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
123. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
124. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x / x2 = 0,
125. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x / x2 = 0, lim x/x = 1,
126. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x / x2 = 0, lim x/x = 1, and lim x2/x = ∞.
127. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x / x2 = 0, lim x/x = 1, and lim x2/x = ∞.
Again, all these questions are in the form ∞/∞
but have different behaviors as x ∞.
128. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
Both questions are in the form of ∞ – ∞ yet they
have drastically different behaviors as x ∞.
2. ∞ / ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,
but lim x / x2 = 0, lim x/x = 1, and lim x2/x = ∞.
Again, all these questions are in the form ∞/∞
but have different behaviors as x ∞.
We have to find other ways to determine the
limiting behaviors when a problem is in the
inconclusive ∞ – ∞ and ∞ / ∞ form.
129. Limits I
3x + 4
5x + 6
For example the is of the ∞ / ∞ form
as x ∞, therefore we will have to transform the
formula to determine its behavior.
130. Limits I
3x + 4
5x + 6
For example the is of the ∞ / ∞ form
as x ∞, therefore we will have to transform the
formula to determine its behavior.
We will talk about various methods in the next
section in determining the limits of formulas with
inconclusive forms and see that
3x + 4
5x + 6
lim = 3/5. ∞
(Take out the calculator and try to find it.)