The document provides an introduction to evaluating limits, including:
1. The limit of a constant function is the constant.
2. Common limit laws can be used to evaluate limits of sums, differences, products, and quotients if the individual limits exist.
3. Special techniques may be needed to evaluate limits that involve indeterminate forms, such as 0/0, infinity/infinity, or limits approaching infinity. These include factoring, graphing, and rationalizing.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Day 3 of Free Intuitive Calculus Course: Limits by FactoringPablo Antuna
Today we focus on limits by factoring. We solve limits by factoring and cancelling. This is one of the basic techniques for solving limits. We talk about the idea behind this technique and we solve some examples step by step.
Day 5 of the Intuitive Online Calculus Course: The Squeeze TheoremPablo Antuna
In this presentation we learn about the Squeeze Theorem. We first try to get the intuition behind it, why it must be true. Then we apply it to solve the Fundamental Trigonometric Limit. This limit is very important for solving other trigonometric limits.
To solve this limit we use a little bit of geometry and then apply the Squeeze Theorem.
The concept of limit formalizes the notion of closeness of the function values to a certain value "near" a certain point. Limits behave well with respect to arithmetic--usually. Division by zero is always a problem, and we can't make conclusions about nonexistent limits!
In this second day we solve the most basic limits we could find, like the limit of a constant. Then we find the limit of the sum, the product and the quotient of two functions. We solve two simple examples.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
We solve limits by rationalizing. This is the second technique you may learn after limits by factoring. We solve two examples step by step.
Watch video: http://www.youtube.com/watch?v=8CtpuojMJzA
More videos and lessons: http://www.intuitive-calculus.com/solving-limits.html
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Day 3 of Free Intuitive Calculus Course: Limits by FactoringPablo Antuna
Today we focus on limits by factoring. We solve limits by factoring and cancelling. This is one of the basic techniques for solving limits. We talk about the idea behind this technique and we solve some examples step by step.
Day 5 of the Intuitive Online Calculus Course: The Squeeze TheoremPablo Antuna
In this presentation we learn about the Squeeze Theorem. We first try to get the intuition behind it, why it must be true. Then we apply it to solve the Fundamental Trigonometric Limit. This limit is very important for solving other trigonometric limits.
To solve this limit we use a little bit of geometry and then apply the Squeeze Theorem.
The concept of limit formalizes the notion of closeness of the function values to a certain value "near" a certain point. Limits behave well with respect to arithmetic--usually. Division by zero is always a problem, and we can't make conclusions about nonexistent limits!
In this second day we solve the most basic limits we could find, like the limit of a constant. Then we find the limit of the sum, the product and the quotient of two functions. We solve two simple examples.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
We solve limits by rationalizing. This is the second technique you may learn after limits by factoring. We solve two examples step by step.
Watch video: http://www.youtube.com/watch?v=8CtpuojMJzA
More videos and lessons: http://www.intuitive-calculus.com/solving-limits.html
Attentes des jeunes tunisiens des hommes politiques et des médiasNouha Belaid
Ce sondage a été présenté dans le cadre de la journée d'étude " Médias et Communication Politique en Tunisie".
Equipe: https://compolitiquetn.wordpress.com/equipe/
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. 12.2 Limit Theorems
Proverbs 1:7. “The fear of the Lord is the beginning of
knowledge, but fools despise wisdom and instruction.”
2. If f(x) is equal to a constant, k,
then lim f ( x ) = k
x→c
3. If f(x) is equal to a constant, k,
then lim f ( x ) = k
x→c
The limit of a constant is that constant.
4. If f(x) is equal to a constant, k,
then lim f ( x ) = k
x→c
The limit of a constant is that constant.
If f ( x ) = 4, evaluate lim f ( x )
x→1
5. If f(x) is equal to a constant, k,
then lim f ( x ) = k
x→c
The limit of a constant is that constant.
If f ( x ) = 4, evaluate lim f ( x )
x→1
4
6. If f(x) is equal to a constant, k,
then lim f ( x ) = k
x→c
The limit of a constant is that constant.
If f ( x ) = 4, evaluate lim f ( x )
x→1
4
(sketch and show graphically)
7. m
If f ( x ) = x , m ∈+° ,
m m
then lim x = c
x→c
8. m
If f ( x ) = x , m ∈+° ,
m m
then lim x = c
x→c
Always try to evaluate limits by using
substitution first!
9. m
If f ( x ) = x , m ∈+° ,
m m
then lim x = c
x→c
Always try to evaluate limits by using
substitution first!
3 3
lim x = 2 = 8
x→2
10. m
If f ( x ) = x , m ∈+° ,
m m
then lim x = c
x→c
Always try to evaluate limits by using
substitution first!
3 3
lim x = 2 = 8
x→2
(sketch and show graphically)
11. If lim f ( x ) = L and lim g ( x ) = M both exist, then
x→c x→c
12. If lim f ( x ) = L and lim g ( x ) = M both exist, then
x→c x→c
1. lim ⎡ f ( x ) + g ( x ) ⎤ = lim f ( x ) + lim g ( x )
⎣ ⎦ x→c
x→c x→c
13. If lim f ( x ) = L and lim g ( x ) = M both exist, then
x→c x→c
1. lim ⎡ f ( x ) + g ( x ) ⎤ = lim f ( x ) + lim g ( x )
⎣ ⎦ x→c
x→c x→c
The limit of the sum is the sum of the limits
14. If lim f ( x ) = L and lim g ( x ) = M both exist, then
x→c x→c
1. lim ⎡ f ( x ) + g ( x ) ⎤ = lim f ( x ) + lim g ( x )
⎣ ⎦ x→c
x→c x→c
The limit of the sum is the sum of the limits
2. lim ⎡ f ( x ) − g ( x ) ⎤ = lim f ( x ) − lim g ( x )
⎣ ⎦ x→c
x→c x→c
15. If lim f ( x ) = L and lim g ( x ) = M both exist, then
x→c x→c
1. lim ⎡ f ( x ) + g ( x ) ⎤ = lim f ( x ) + lim g ( x )
⎣ ⎦ x→c
x→c x→c
The limit of the sum is the sum of the limits
2. lim ⎡ f ( x ) − g ( x ) ⎤ = lim f ( x ) − lim g ( x )
⎣ ⎦ x→c
x→c x→c
3. lim ⎡ f ( x ) ⋅ g ( x ) ⎤ = lim f ( x ) ⋅ lim g ( x )
⎣ ⎦ x→c
x→c x→c
16. If lim f ( x ) = L and lim g ( x ) = M both exist, then
x→c x→c
1. lim ⎡ f ( x ) + g ( x ) ⎤ = lim f ( x ) + lim g ( x )
⎣ ⎦ x→c
x→c x→c
The limit of the sum is the sum of the limits
2. lim ⎡ f ( x ) − g ( x ) ⎤ = lim f ( x ) − lim g ( x )
⎣ ⎦ x→c
x→c x→c
3. lim ⎡ f ( x ) ⋅ g ( x ) ⎤ = lim f ( x ) ⋅ lim g ( x )
⎣ ⎦ x→c
x→c x→c
f ( x ) lim f ( x )
4. lim = x→c
, g ( x ) & lim g ( x ) ≠ 0
x→c g ( x ) lim g ( x ) x→c
x→c
25. Find each limit:
3
x − 4x Rats! Can’t use substitution as
4. lim 2
x→−1 x + x we get zero in the denominator.
26. Find each limit:
3
x − 4x Rats! Can’t use substitution as
4. lim 2
x→−1 x + x we get zero in the denominator.
x(x − 4)
2
lim
x→−1 x ( x + 1)
27. Find each limit:
3
x − 4x Rats! Can’t use substitution as
4. lim 2
x→−1 x + x we get zero in the denominator.
x(x − 4)
2
lim
x→−1 x ( x + 1)
2
x −4
lim
x→−1 x + 1
28. Find each limit:
3
x − 4x Rats! Can’t use substitution as
4. lim 2
x→−1 x + x we get zero in the denominator.
x(x − 4)
2
lim
x→−1 x ( x + 1)
x −42 Nice try ... but still have zero in
lim the denominator. Check out the
x→−1 x + 1
graph of the original function.
29. Find each limit:
3
x − 4x Rats! Can’t use substitution as
4. lim 2
x→−1 x + x we get zero in the denominator.
x(x − 4)
2
lim
x→−1 x ( x + 1)
x −42 Nice try ... but still have zero in
lim the denominator. Check out the
x→−1 x + 1
graph of the original function.
The limit does not exist!
31. Find each limit:
2
x − 36 Again ... can’t use substitution.
5. lim
x→−6 x + 6 Let’s try factoring again.
32. Find each limit:
2
x − 36 Again ... can’t use substitution.
5. lim
x→−6 x + 6 Let’s try factoring again.
lim
( x − 6 )( x + 6 )
x→−6 ( x + 6)
33. Find each limit:
2
x − 36 Again ... can’t use substitution.
5. lim
x→−6 x + 6 Let’s try factoring again.
lim
( x − 6 )( x + 6 )
x→−6 ( x + 6)
lim ( x − 6 )
x→−6
34. Find each limit:
2
x − 36 Again ... can’t use substitution.
5. lim
x→−6 x + 6 Let’s try factoring again.
lim
( x − 6 )( x + 6 )
x→−6 ( x + 6)
lim ( x − 6 )
x→−6
−12
35. Find each limit:
2
x − 36 Again ... can’t use substitution.
5. lim
x→−6 x + 6 Let’s try factoring again.
lim
( x − 6 )( x + 6 )
x→−6 ( x + 6)
lim ( x − 6 )
x→−6
−12 Verify graphically
36. Find each limit:
2
x − 36 Again ... can’t use substitution.
5. lim
x→−6 x + 6 Let’s try factoring again.
lim
( x − 6 )( x + 6 )
x→−6 ( x + 6)
lim ( x − 6 )
x→−6
−12 Verify graphically
If substitution can’t be used, try to manipulate the
function until substitution will work.
38. Find each limit:
x−3
6. lim 2
x→3 x − 9
x−3
lim
x→3 ( x + 3) ( x − 3)
39. Find each limit:
x−3
6. lim 2
x→3 x − 9
x−3
lim
x→3 ( x + 3) ( x − 3)
1
6
40. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
41. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
2
x + 13x
7. lim
x→∞ 2x 2 − 5
42. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
2
x + 13x Substitution yields infinity over
7. lim infinity which is indeterminable.
x→∞ 2x 2 − 5
43. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
2
x + 13x Substitution yields infinity over
7. lim infinity which is indeterminable.
x→∞ 2x 2 − 5
1
2
Multiply by: x
1
2
x
44. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
2
x + 13x Substitution yields infinity over
7. lim infinity which is indeterminable.
x→∞ 2x 2 − 5
1
2
13 Multiply by: x
1+ 1
lim x
x→∞ 5 x 2
2− 2
x
45. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
2
x + 13x Substitution yields infinity over
7. lim infinity which is indeterminable.
x→∞ 2x 2 − 5
1
2
13 Multiply by: x
1+ 1
lim x
x→∞ 5 x 2
2− 2
x
k
Recall: lim =0
x→∞ x
46. Limits approaching infinity require their own
unique techniques. Let’s get an introduction
to these techniques.
2
x + 13x Substitution yields infinity over
7. lim infinity which is indeterminable.
x→∞ 2x 2 − 5
1
2
13 Multiply by: x
1+ 1
lim x
x→∞ 5 x 2
2− 2
x
k
1 Recall: lim =0
x→∞ x
2