# Evaluate the limit for the following if exists. $\;\lim\limits_{x \to \infty}\large\frac{\sin\frac{2}{x}}{\Large\frac{1}{x}}$

Toolbox:
• L'Hopital's rule: Let $f$ and $g$ be continous real valued functions defined on the closed interval $[a,b], f,g$ be differentiable on $(a,b)$ and $g'(c) \neq 0$
• Then if $\lim\limits_{x \to c}\; f(x)=0, \lim \limits_{x \to c}\; g(x)=0$ and
• $\lim\limits_{x \to c} \large\frac{f'(x)}{g'(x)}$$=L it follows that • \lim \limits_{x \to c} \large\frac{f(x)}{g(x)}$$=L$
Step 1:
$\;\lim\limits_{x \to \infty} \large\frac{\sin \Large\frac{2}{x}}{\frac{1}{x}}$
Let $\large\frac{1}{x}$$=y As x \to \infty ,y \to 0 Step 2: \lim \limits_{x \to \infty}\; \large\frac{\sin \Large\frac{2}{x}}{\Large\frac{1}{x}}$$=\lim\limits _{y \to 0}\; \large\frac{\sin 2y}{y}$ Which is of the form $\large\frac{0}{0}$
Step 3:
Applying L'Hopital's rule,
$\lim \limits_{x \to \infty}\; \large\frac{\sin \Large\frac{2}{x}}{\Large\frac{1}{x}}$$=\lim\limits _{y \to 0}\; \Large\frac{2 \cos 2y}{1}$$=2$

edited Jul 29, 2013 by meena.p