Chapter 14.1, Problem 16AYU

To determine

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

Answer to Problem 16AYU

Solution:

$1$

Explanation of Solution

Given:

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

Calculation:

$\underset{x\to 0}{\lim }\frac{\tan x}{x}=\underset{x\to 0}{\lim }\frac{\left(\frac{\sin x}{\cos x}\right)}{x}=\underset{x\to 0}{\lim }\left(\frac{\sin x}{x}\times \frac{1}{\cos x}\right)=\underset{x\to 0}{\lim }\left(\frac{\sin x}{x}\right)\times \underset{x\to 0}{\text{lim}}\left(\frac{1}{\cos x}\right)=1\times \left(\frac{1}{\cos 0}\right)=1\times 1=1$