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Evaluate the limit for the following if exists. $\;\lim\limits_{x \to 0} \large\frac{\cot x}{\cot 2x}$

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  • 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$
$\;\lim\limits_{x \to 0} \large\frac{\cot x}{\cot 2x}$ is of the form $\large\frac{\infty}{\infty}$
This can be rewritten as $ \lim \limits_{x \to 0} \large\frac{\Large\frac{1}{\cot 2x}}{\Large\frac{1}{\cot x}}$
$\;\lim\limits_{x \to 0} \large\frac{\tan 2x}{\tan x}$ which is of the form $\large\frac{0}{0}$
Step 2:
We have, by applying L'Hopital's rule
$\;\lim\limits_{x \to 0} \large\frac{\cot x}{\cot 2x}$$=\lim\limits_{x \to 0}\large\frac{2 \sec ^2 2x}{\sec ^2 x}$$=2$


answered Jul 29, 2013 by meena.p
edited Jul 29, 2013 by meena.p

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