Q. 32 L鈥橦藛opital鈥檚 Rule limit cal... [FREE SOLUTION]

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Chapter 3: Q. 32 (page 314)

L鈥橦藛opital鈥檚 Rule limit calculations: Calculate each of the
limits that follow. Some of these limits are easier to calculate
by using L鈥橦opital鈥檚 rule, and some are not.
$\lim_{x 鈫� 0^{+}} x^{1 - \cos x}$

Short Answer

Expert verified

Limit is $1$ and it is easy to find using L'Hospital rule

Step by step solution

Step 1. Given information

An expression is given as $\lim_{x 鈫� 0^{+}} x^{1 - \cos x}$

Step 2. Evaluating limit

Use properties of limits as,

$\lim_{x 鈫� 0^{+}} \ln x^{1 - \cos x} = \lim_{x 鈫� 0^{+}} (1 - \cos x) \ln x = \lim_{x 鈫� 0^{+}} \frac{\ln x}{(1 - \cos x)} = \lim_{x 鈫� 0^{+}} \frac{\frac{1}{x}}{- \frac{(- (- \sin x))}{(1 - \cos x)^{2}}} = - \lim_{x 鈫� 0^{+}} \frac{(1 - \cos x)^{2}}{x \sin x} = - \lim_{x 鈫� 0^{+}} \frac{1 - 2 \cos x + \cos^{2} x}{x \sin x} = - \lim_{x 鈫� 0^{+}} \frac{2 \sin x + 2 \cos x (- \sin x)}{x \cos x + \sin x 脳 1} = - 2 \lim_{x 鈫� 0^{+}} \frac{\sin x - \sin x \cos x}{x \cos x + \sin x} = - 2 \lim_{x 鈫� 0^{+}} \frac{\cos x - \sin x (- \sin x) - \cos x \cos x}{x (- \sin x) + \cos x + \cos x} = - 2 \lim_{x 鈫� 0^{+}} \frac{\cos x - \sin^{2} x - \cos^{2} x}{- x \sin x + 2 \cos x} = - 2 \lim_{x 鈫� 0^{+}} \frac{\cos x - (\sin^{2} x + \cos^{2} x)}{- x \sin x + 2 \cos x} = - 2 \lim_{x 鈫� 0^{+}} \frac{\cos x - 1}{- x \sin x + 2 \cos x} = - 2 \frac{\cos 0 - 1}{- 0 \sin 0 + 2 \cos 0} = - 2 脳 \frac{1 - 1}{0 + 2 脳 1} = - 2 脳 \frac{0}{2} = 0 \underset{x 鈫� 0^{+}}{鈬� \lim} x^{1 - \cos x} = e^{0} = 1$

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91影视

L鈥橦藛opital鈥檚 Rule limit calculations: Calculate each of the
limits that follow. Some of these limits are easier to calculate
by using L鈥橦opital鈥檚 rule, and some are not.
$\lim_{x 鈫� 0^{+}} x^{1 - \cos x}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating limit

One App. One Place for Learning.

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91影视

L鈥橦藛opital鈥檚 Rule limit calculations: Calculate each of thelimits that follow. Some of these limits are easier to calculateby using L鈥橦opital鈥檚 rule, and some are not.limx鈫�0+x1-cosx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Logic and Functions

Mechanics Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

L鈥橦藛opital鈥檚 Rule limit calculations: Calculate each of the
limits that follow. Some of these limits are easier to calculate
by using L鈥橦opital鈥檚 rule, and some are not.
$\lim_{x 鈫� 0^{+}} x^{1 - \cos x}$