Chapter 9.2, Problem 64E

To determine

To find : The value of $\underset{x\to 0}{\lim }\frac{x}{\tan x}$

Answer to Problem 64E

  $\underset{x\to 0}{\lim }\frac{x}{\tan x}=1$

Answer (C).

Explanation of Solution

Given information : $\underset{x\to 0}{\lim }\frac{x}{\tan x}$

Formula used : L’Hospital Rule.

Calculation :

  $\begin{array}{l}\underset{x\to 0}{\lim }\frac{x}{\tan x}\left[\equiv \frac{0}{0}\right]\\ =\underset{x\to 0}{\lim }\frac{x}{{\sec }^{2}x}\\ =\underset{x\to 0}{\lim }{\cos }^{2}x=1\end{array}$

Therefore, $\underset{x\to 0}{\lim }\frac{x}{\tan x}=1$