2.1 Q-13

Q13. ${\lim }_{x\to 0}\frac{x}{\sin 3x}$

\section*{Solution:} $\begin{array}{rl}& \underset{x\to 0}{\lim }\frac{x}{\sin 3x}\\ & =\underset{x\to 0}{\lim }\frac{x}{\frac{\sin 3x}{3x}\times 3x}\\ & =\underset{x\to 0}{\lim }\frac{1}{3\cdot \sin \frac{3x}{3x}}\\ & =\frac{1}{3}\cdot \frac{1}{{\lim }_{x\to 0}\frac{\sin 3x}{3x}}\\ & =\frac{1}{3}\cdot \frac{1}{1}=\frac{1}{3}\end{array}$