This document provides an overview of L'Hopital's rule, which can be used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It begins by defining indeterminate limits and the different types that occur, including 0/0, ∞/∞, ∞⋅0, ∞-∞, and ∞0. It then states L'Hopital's rule, which allows such limits to be evaluated by taking the derivative of the numerator and denominator. The document provides examples of applying the rule to various indeterminate forms and discusses common mistakes. It also notes other strategies like rewriting expressions as quotients before applying the rule.
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.
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.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
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Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
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Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
1. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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31. L’Hopital’s Rule
31.1. Limit of indeterminate type
Some limits for which the substitution rule does not apply can be found by using inspection.
For instance,
lim
x→0
cos x
x2
about 1
small pos.
= ∞
On the other hand, we have seen (8) that inspection cannot be used to find the limit of a
fraction when both numerator and denominator go to 0. The examples given were
lim
x→0+
x2
x
, lim
x→0+
x
x2
, lim
x→0+
x
x
.
In each case, both numerator and denominator go to 0. If we had a way to use inspection to
decide the limit in this case, then it would have to give the same answer in all three cases.
Yet, the first limit is 0, the second is ∞ and the third is 1 (as can be seen by canceling x’s).
We say that each of the above limits is indeterminate of type 0
0 . A useful way to
remember that one cannot use inspection in this case is to imagine that the numerator
going to 0 is trying to make the fraction small, while the denominator going to 0 is trying
to make the fraction large. There is a struggle going on. In the first case above, the
numerator wins (limit is 0); in the second case, the denominator wins (limit is ∞); in the
third case, there is a compromise (limit is 1).
Changing the limits above so that x goes to infinity instead gives a different indeterminate
2. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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type. In each of the limits
lim
x→∞
x2
x
, lim
x→∞
x
x2
, lim
x→∞
x
x
.
both numerator and denominator go to infinity. The numerator going to infinity is trying
to make the fraction large, while the denominator going to infinity is trying to make the
fraction small. Again, there is a struggle. Once again, we can cancel x’s to see that the
first limit is ∞ (numerator wins), the second limit is 0 (denominator wins), and the third
limit is 1 (compromise). The different answers show that one cannot use inspection in this
case. Each of these limits is indeterminate of type ∞
∞ .
Sometimes limits of indeterminate types 0
0 or ∞
∞ can be determined by using some algebraic
technique, like canceling between numerator and denominator as we did above (see also
12). Usually, though, no such algebraic technique suggests itself, as is the case for the limit
lim
x→0
x2
sin x
,
which is indeterminate of type 0
0 . Fortunately, there is a general rule that can be applied,
namely, l’Hôpital’s rule.
3. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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31.2. L’Hôpital’s rule
L’Hôpital’s rule. If the limit
lim
f(x)
g(x)
is of indeterminate type 0
0 or ±∞
±∞ , then
lim
f(x)
g(x)
= lim
f0
(x)
g0(x)
,
provided this last limit exists. Here, lim stands for lim
x→a
, lim
x→a±
, or lim
x→±∞
.
The pronunciation is lō-pē-täl. Evidently, this result is actually due to the mathematician
Bernoulli rather than to l’Hôpital. The verification of l’Hôpital’s rule (omitted) depends
on the mean value theorem.
31.2.1 Example Find lim
x→0
x2
sin x
.
Solution As observed above, this limit is of indeterminate type 0
0 , so l’Hôpital’s rule
applies. We have
lim
x→0
x2
sin x
0
0
l’H
= lim
x→0
2x
cos x
=
0
1
= 0,
where we have first used l’Hôpital’s rule and then the substitution rule.
4. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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The solution of the previous example shows the notation we use to indicate the type of an
indeterminate limit and the subsequent use of l’Hôpital’s rule.
31.2.2 Example Find lim
x→−∞
3x − 2
ex2 .
Solution We have
lim
x→−∞
3x − 2
ex2
−∞
∞
l’H
= lim
x→−∞
3
ex2
(2x)
3
large neg.
= 0.
31.3. Common mistakes
Here are two pitfalls to avoid:
ˆ L’Hôpital’s rule should not be used if the limit is not indeterminate (of the appropriate
type). For instance, the following limit is not indeterminate; in fact, the substitution
rule applies to give the limit:
lim
x→0
sin x
x + 1
=
0
1
= 0.
An application of l’Hôpital’s rule gives the wrong answer:
lim
x→0
sin x
x + 1
l’H
= lim
x→0
cos x
1
=
1
1
= 1 (wrong).
5. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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ˆ Although l’Hôpital’s rule involves a quotient f(x)/g(x) as well as derivatives, the
quotient rule of differentiation is not involved. The expression in l’Hôpital’s rule is
f0
(x)
g0(x)
and not
f(x)
g(x)
0
.
31.4. Examples
31.4.1 Example Find lim
θ→0
sin θ
θ
.
Solution We have
lim
θ→0
sin θ
θ
0
0
l’H
= lim
θ→0
cos θ
1
=
1
1
= 1.
(In 19, we had to work pretty hard to determine this important limit. It is tempting to go
back and replace that argument with this much easier one, but unfortunately we used this
limit to derive the formula for the derivative of sin θ, which is used here in the application
of l’Hôpital’s rule, so that would make for a circular argument.)
Sometimes repeated use of l’Hôpital’s rule is called for:
31.4.2 Example Find lim
x→∞
3x2
+ x + 4
5x2 + 8x
.
Solution We have
lim
x→∞
3x2
+ x + 4
5x2 + 8x
∞
∞
l’H
= lim
x→∞
6x + 1
10x + 8
∞
∞
l’H
= lim
x→∞
6
10
=
6
10
=
3
5
.
6. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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For the limit at infinity of a rational function (i.e., polynomial over polynomial) as in the
preceding example, we also have the method of dividing numerator and denominator by
the highest power of the variable in the denominator (see 12). That method is probably
preferable to using l’Hôpital’s rule repeatedly, especially if the degrees of the polynomials
are large. Sometimes though, we have no alternate approach:
31.4.3 Example Find lim
x→0
ex
− 1 − x − x2
/2
x3
.
Solution We have
lim
x→0
ex
− 1 − x − x2
/2
x3
0
0
l’H
= lim
x→0
ex
− 1 − x
3x2
0
0
l’H
= lim
x→0
ex
− 1
6x
0
0
l’H
= lim
x→0
ex
6
=
e0
6
=
1
6
.
There are other indeterminate types, to which we now turn. The strategy for each is to
transform the limit into either type 0
0 or ±∞
±∞ and then use l’Hôpital’s rule.
7. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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31.5. Indeterminate product
Type ∞ · 0. The limit
lim
x→∞
xe−x
(∞ · 0)
cannot be determined by using inspection. The first factor going to ∞ is trying to make
the expression large, while the second factor going to 0 is trying to make the expression
small. There is a struggle going on. We say that this limit is indeterminate of type ∞ · 0.
The strategy for handling this type is to rewrite the product as a quotient and then use
l’Hôpital’s rule.
31.5.1 Example Find lim
x→∞
xe−x
.
Solution As noted above, this limit is indeterminate of type ∞ · 0. We rewrite the expres-
sion as a fraction and then use l’Hôpital’s rule:
lim
x→∞
xe−x
= lim
x→∞
x
ex
∞
∞
l’H
= lim
x→∞
1
ex
1
large pos.
= 0.
31.5.2 Example Find lim
x→0+
(cot 2x)(sin 6x).
8. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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Solution First
lim
x→0+
cot 2x = lim
x→0+
cos 2x
sin 2x
about 1
small pos.
= ∞.
Therefore, the given limit is indeterminate of type ∞ · 0. We rewrite as a fraction and then
use l’Hôpital’s rule:
lim
x→0+
(cot 2x)(sin 6x) = lim
x→0+
sin 6x
tan 2x
0
0
l’H
= lim
x→0+
6 cos 6x
2 sec2 2x
=
6
2
= 3.
31.6. Indeterminate difference
Type ∞ − ∞. The substitution rule cannot be used on the limit
lim
x→ π
2
−
(tan x − sec x) (∞ − ∞)
since tan π/2 is undefined. Nor can one determine this limit by using inspection. The first
term going to infinity is trying to make the expression large and positive, while the second
term going to negative infinity is trying to make the expression large and negative. There
is a struggle going on. We say that this limit is indeterminate of type ∞ − ∞.
9. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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The strategy for handling this type is to combine the terms into a single fraction and then
use l’Hôpital’s rule.
31.6.1 Example Find lim
x→ π
2
−
(tan x − sec x).
Solution As observed above, this limit is indeterminate of type ∞ − ∞. We combine the
terms and then use l’Hôpital’s rule:
lim
x→ π
2
−
(tan x − sec x) = lim
x→ π
2
−
sin x
cos x
−
1
cos x
= lim
x→ π
2
−
sin x − 1
cos x
0
0
l’H
= lim
x→ π
2
−
cos x
− sin x
=
0
−1
= 0.
31.6.2 Example Find lim
x→1+
x
x − 1
−
1
ln x
.
Solution The limit is indeterminate of type ∞ − ∞. We combine the terms and then use
10. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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l’Hôpital’s rule:
lim
x→1+
x
x − 1
−
1
ln x
= lim
x→1+
x ln x − x + 1
(x − 1) ln x
0
0
l’H
= lim
x→1+
ln x + x(1/x) − 1
ln x + (x − 1)(1/x)
= lim
x→1+
ln x
ln x + 1 − 1/x
0
0
l’H
= lim
x→1+
1/x
1/x + 1/x2
=
1
2
.
31.7. Indeterminate powers
Type ∞0
. The limit
lim
x→∞
x1/x
(∞0
)
cannot be determined by using inspection. The base going to infinity is trying to make the
expression large, while the exponent going to 0 is trying to make the expression equal to
1. There is a struggle going on. We say that this limit is indeterminate of type ∞0
.
The strategy for handling this type (as well as the types 1∞
and 00
yet to be introduced) is
to first find the limit of the natural logarithm of the expression (ultimately using l’Hôpital’s
rule) and then use an inverse property of logarithms to get the original limit.
11. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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31.7.1 Example Find lim
x→∞
x1/x
.
Solution As was noted above, this limit is of indeterminate type ∞0
. We first find the
limit of the natural logarithm of the given expression:
lim
x→∞
ln x1/x
= lim
x→∞
(1/x) ln x (0 · ∞)
= lim
x→∞
ln x
x
∞
∞
l’H
= lim
x→∞
1/x
1
= 0.
Therefore,
lim
x→∞
x1/x
= lim
x→∞
eln x1/x
= e0
= 1,
where we have used the inverse property of logarithms y = eln y
and then the previous com-
putation. (The next to the last equality also uses continuity of the exponential function.)
Type 1∞
. The limit
lim
x→∞
1 +
1
x
x
(1∞
)
cannot be determined by using inspection. The base going to 1 is trying to make the
expression equal to 1, while the exponent going to infinity is trying to make the expression
go to ∞ (raising a number greater than 1 to ever higher powers produces ever larger results).
We say that this limit is indeterminate of type 1∞
.
12. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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31.7.2 Example Find lim
x→∞
1 +
1
x
x
.
Solution As was noted above, this limit is indeterminate of type 1∞
. We first find the
limit of the natural logarithm of the given expression:
lim
x→∞
ln
1 +
1
x
x
= lim
x→∞
x ln
1 +
1
x
= lim
x→∞
ln (1 + 1/x)
x−1
0
0
l’H
= lim
x→∞
1
1 + 1/x
−x−2
−x−2
= lim
x→∞
1
1 + 1/x
= 1.
Therefore,
lim
x→∞
1 +
1
x
x
= lim
x→∞
eln(1+ 1
x )
x
= e1
= e.
Type 00
. The limit
lim
x→0
(tan x)x
00
cannot be determined by using inspection. The base going to 0 is trying to make the
expression small, while the exponent going to 0 is trying to make the expression equal to
1. We say that this limit is indeterminate of type 00
.
13. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
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31.7.3 Example Find lim
x→0+
(tan x)x
.
Solution As was noted above, this limit is indeterminate of type 00
. We first find the limit
of the natural logarithm of the given expression:
lim
x→0+
ln(tan x)x
= lim
x→0+
x ln tan x
= lim
x→0+
ln tan x
x−1
−∞
∞
l’H
= lim
x→0+
cot x sec2
x
−x−2
= lim
x→0+
−x2
sin x cos x
0
0
l’H
= lim
x→0+
−2x
cos2 x − sin2
x
=
0
1
= 0.
Therefore,
lim
x→0+
(tan x)x
= lim
x→0+
eln(tan x)x
= e0
= 1.
Between the two applications of l’Hôpital’s rule in the last example, we used algebra to
simplify the expression. Indiscriminate repeated use of l’Hôpital’s rule with no attempt at
simplification can give rise to unwieldy expressions, so it is best to make the expression as
simple as possible before each application of the rule.
14. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
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31.8. Summary
We summarize the indeterminate types:
Indeterminate types.
0
0
,
±∞
±∞
, ±∞ · 0, ∞ − ∞, ∞0
, 1±∞
, 00
.
Limits with the following forms are not indeterminate since the parts work together toward
a common goal (no struggle):
0
∞
,
∞
0
, ∞ · ∞, ∞ + ∞, ∞∞
, 0∞
.
The last of these, 0∞
, might require some explanation. The base is approaching 0, so it
can be made close enough to 0 that it is at least less than 1 in absolute value. But raising
a number less than 1 in absolute value to ever higher powers produces numbers ever closer
to 0 (for instance, 1
2 raised to the powers 1, 2, 3, . . . produces 1
2 , 1
4 , 1
8 , . . .). Therefore, both
base and exponent work together to produce a limit of 0.
Due to the great variety of forms limits can take on, it is recommended that the reader
employ the sort of imagery we have used (of parts either struggling against each other or
working together toward a common goal) as an aid to keeping straight which forms are
indeterminate and which are not.
31.8.1 Example Find lim
x→0+
(1 − ex
)
1/x
.
15. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
JJ II
J I
Page 15 of 17
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Solution This limit has the form 0∞
, which is not indeterminate (both base and exponent
are working together to make the expression small). The limit is 0.
16. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
JJ II
J I
Page 16 of 17
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31 – Exercises
31 – 1 Find lim
x→0
e3x
− 1
x
.
31 – 2 Find lim
x→∞
ln x
x2
.
31 – 3 Find lim
x→0+
x ln x.
31 – 4 Find lim
x→0
ex
− x − 1
cos x − 1
.
31 – 5 Find lim
x→0+
1
sin x
−
1
x
.
31 – 6 Find lim
x→π/2
1 − sin x
csc x
.
17. L’Hopital’s Rule
Limit of indeterminate type
L’Hôpital’s rule
Common mistakes
Examples
Indeterminate product
Indeterminate difference
Indeterminate powers
Summary
Table of Contents
JJ II
J I
Page 17 of 17
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31 – 7 Find lim
x→0+
(sin x)x
.
31 – 8 Find lim
x→∞
(ex
+ x)
1/x
.