Question

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.

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

Short answer

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

Step-by-step solution

Step 1. Given information

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

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

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

$$\begin{gathered} =\underset{x\to 0}{\lim }\frac{1}{\left(1+x^2\right)\cos x} \\ =\frac{1}{\left(1+0^2\right)\cos 0} \\ =\frac{1}{1} \\ =1 \end{gathered}$$

Therefore, the exact value is,$1.$