2.1 Q-12

Q12. ${\lim }_{x\to 0}\frac{\tan x}{x}$

\section*{Solution:} $\begin{array}{rl}& \underset{x\to 0}{\lim }\frac{\tan x}{x}\\ & =\underset{x\to 0}{\lim }\frac{\frac{\sin x}{\cos x}}{x}\\ & =\underset{x\to 0}{\lim }\frac{\sin x}{x\cdot \cos x}\\ & =\underset{x\to 0}{\lim }\frac{\sin x}{x}\cdot \underset{x\to 0}{\lim }\frac{1}{\cos x}\\ & =1\cdot \frac{1}{\cos (0)}\\ & =1\cdot \frac{1}{1}\\ & =1\end{array}$