Question

Evaluate :


$\displaystyle\lim_{x \to 0}\: {(Cos x)}^{\frac{1}{x^2}}$

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Answer 1

Answer:

Step-by-step explanation:

Answer 2

ANSWER

$e^{\frac{-1}{2} }$

Step By Step Explanation

$\displaystyle\lim_{x \to 0}\: {(Cos x)}^{\frac{1}{x^2}}$

Here we know that >>

It is of $(\rightarrow 1)^{ \rightarrow\infty}$ form.

Now we get ->

$\implies e^{ \displaystyle \lim_{x \to 0}\: \left(cos x - 1)(\dfrac{1}{x^{2} }\right) }$

$\implies e^{ \displaystyle \lim_{x \to 0}\: \dfrac{-2Sin^{2} \dfrac{x}{2} }{x^{2}}}$

$\implies e^{ \displaystyle \lim_{x \to 0}\: \dfrac{-2Sin^{2} \dfrac{x}{2} }{4.\left(\dfrac{x}{2}\right)^{2}}}$

$\implies e^{ \displaystyle \lim_{x \to 0}\: \dfrac{1}{2}\left(\dfrac{Sin\dfrac{x}{2} }{\dfrac{x}{2}} \right)^{2}}$

$\implies e^{\frac{-1}{2} \times 1}$

$\implies e^{\frac{-1}{2} }$

Hence,on evaluating $\displaystyle \lim_{x \to 0}\: {(Cos x)}^{\frac{1}{x^2}}$ we get $e^{\frac{-1}{2} }$.