Chapter 3: Q. 47 (page 311)

Calculate each of the limits in Exercises $21 - 48$ . Some of these limits are made easier by L’Hopital’s rule, and some are not.
$\lim_{x \to 0} \frac{\tan^{- 1} x}{\sin x}$ .

Short Answer

Expert verified

The exact value of the limit $\lim_{x \to 0} \frac{\tan^{- 1} x}{\sin x}$ is, $1 .$

Step by step solution

Step 1. Given information

$\lim_{x \to 0} \frac{\tan^{- 1} x}{\sin x}$ .

Step 2. The limit using L'Hopital's rule is given below:

$\lim_{x \to 0} \frac{\tan^{- 1} x}{\sin x} = \lim_{x \to 0} \frac{\frac{1}{1 + x^{2}}}{\cos x}$ [L'Hopital's rule]

$= \lim_{x \to 0} \frac{1}{(1 + x^{2}) \cos x} = \frac{1}{(1 + 0^{2}) \cos 0} = \frac{1}{1} = 1$

Therefore, the exact value is, $1 .$

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Calculate each of the limits in Exercises $21 - 48$ . Some of these limits are made easier by L’Hopital’s rule, and some are not.
$\lim_{x \to 0} \frac{\tan^{- 1} x}{\sin x}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The limit using L'Hopital's rule is given below:

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Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L’Hopital’s rule, and some are not.limx→0tan-1xsinx.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The limit using L'Hopital's rule is given below:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Calculate each of the limits in Exercises $21 - 48$ . Some of these limits are made easier by L’Hopital’s rule, and some are not.
$\lim_{x \to 0} \frac{\tan^{- 1} x}{\sin x}$ .