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# Evaluate the limit for the following if exists. $\;\lim\limits_{x \to 0} \large\frac{\tan x-x}{x- \sin x}$

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:
$\;\lim\limits_{x \to 0} \large\frac{\tan x-x}{x- \sin x}$ is of the form $\large\frac{0}{0}$
Step 2:
Applying L'Hopital's rule,
$\lim \limits_{x \to 0}\; \large\frac{\tan x- x}{x-\sin x}$$=\lim\limits _{x \to 0}\; \large\frac{\sec^2 x -1}{1-\cos x} Which is again of the form \large\frac{0}{0} Step 3: Applying L'Hopital's rule, once again \lim \limits_{x \to 0}\; \large\frac{2 \sec x \tan x}{\sin x}$$=\lim\limits _{x \to 0}\; 2 \sec^2 x =2$

edited Jul 29, 2013 by meena.p