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MH1100/MTH112: Calculus I.

Tutorial in the final week.

This week’s topics:

•   L’Hospital’s rule.

•   Limits to positive and negative infinity.

•   Basics of antiderivatives and indefinite integrals.

•   Substitution method.

•   Integration by parts.

The problems that will be focussed on will be announced before the tutorial.

Problem 1: (Various from Section 6.8 from [Stewart]).

Below you are asked to determine a number of limits. In you can find amore elementary method than L’Hospital’s rule, please use it instead. If L’Hospital’s rule doesn’t apply, explain why.

1.   limx→1  x2−1x2−x   2.   limx→(π/2)+

cosx1−sinx

3.   limx→0   sin 4xtan5x   4.   limx→∞ lnx√ x

5.   limx→0+ln xx   6.   limt→0

 8t−5tt

7.   limx→0√ 1+2x−

√ 1−4x

x   8.   limx→0  ex−1−x

x2

9.   limx→0 sin−1 x

x   10.   limx→∞(lnx)2

x

11.   limx→1 1−x+ln x1+cosπx   12.   limx→∞ x sin

π

x

13.   limx→0+ sin x ln x   14.   limx→∞ (x− ln x)

15.   limx→0+ x

√ x

16.   limx→∞ x1/x

17⋆.   limx→∞

2x−32x+5

2x+118.   limx→0+

1x −   1ex−1

.

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Problem 2: (#6.8.71 from [Stewart]).

Prove that for any positive integer  n ∈ N:

limx→∞

ex

xn  = ∞.

Problem 3: (#6.8.72 from [Stewart]).

Prove that for any positive real  p ∈ R,  p > 0,

limx→∞

ln xx p

  = 0.

Problem 4⋆: (#6.8.98 from [Stewart]).

For what values of  a  and  b  is the following equation true?

limx→0

sin2x

x3  + a +

  b

x2

 = 0.

Problem 5: (#3.4.8 from [Stewart].)

Evaluate the following limit:

limx→∞

 12x3 − 5x + 21 + 4x2 + 3x3

  .

Problem 6: (#3.4.17 and #3.4.18 from [Stewart].)

Evaluate the limits:

(i) limx→∞√ 9x6−xx3+1

(ii) limx→−∞√ 9x6−xx3+1

  .

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Problem 7: (#3.4.19 from [Stewart].)

Evaluate the limit:limx→∞

 9x2 + x− 3x

.

Problem 8: (#3.4.20 from [Stewart].)

Evaluate the limit:lim

x→−∞

x +

 x2 + 2x

.

Problem 9: (#3.4.21 from [Stewart].)

Evaluate the limit:

limx→∞

 x2 + ax−

 x2 + bx

.

Problem 10: (#3.4.22 from [Stewart].)

Does the limit limx→∞ cos x  exist? Discuss.

Problem 11: (#3.4.27 from [Stewart].)

Evaluate the limitlimx→∞

(x−√ 

x).

Problem 12: (#3.4.29 and #3.4.30 from [Stewart].)

Evaluate the limits

(i) limx→∞ x sin 1x ,

(ii) limx→∞√ 

x sin  1x .

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Problem 13: (#3.4.41 from [Stewart].)

Find a formula for a function f (x) which has all of the following properties:

limx→±∞ f (x) = 0,   limx→0 f (x) = −∞, f (2) = 0,limx→3− f (x) = ∞,   limx→3+ f (x) = −∞.

Problem 14⋆: (#3.4.71 from [Stewart].)

Using the definitions, prove that

limx→∞ f (x) = limt→0+

f  (1/t)

and thatlim

x→−∞f (x) = lim

t→0− f  (1/t) .

(The meaning of these equations is that when either limit exists, then sodoes the other one, and then the given limits are equal.)

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