Question

lim x->0 (cos x)^1/x²​

Answer 1

EXPLANATION.

$\sf \implies \displaystyle \lim_{x \to 0} (cos x)^{1/x^{2} }$

As we know that,

First we put the value of x = 0 in equation and check their indeterminant form, we get.

$\sf \implies \displaystyle \lim_{x \to 0} (cos(0))^{1/0} = \dfrac{\infty}{\infty}$

As we can see that it is a ∞/∞ form indeterminant.

$\sf \implies \displaystyle \lim_{x \to 0} e^{ln \bigg[(cos x)^{1/x^{2} } \bigg]}$

$\sf \implies \displaystyle \lim_{x \to 0} e ^{1/x^{2} \bigg[ ln(cos x) \bigg]}$

$\sf \implies \displaystyle \lim_{x \to 0} e^{\bigg[\dfrac{ln(cos x)}{x^{2} }\bigg] }$

$\sf \implies e^{\displaystyle \lim_{x \to 0} \bigg[\dfrac{ln(cos x)}{x^{2} } \bigg]}$

Put the values of x = 0 in equation again and check indeterminant form, we get.

$\sf \implies e^{\displaystyle \lim_{x \to 0} \bigg[\dfrac{ln(cos(0))}{0} \bigg]$

As we can see that, it is a form of 0/0 indeterminant,

Now apply L-HOSPITAL'S rule, we get.

$\sf \implies e^{\displaystyle \lim_{x \to 0} \bigg[\dfrac{-sin x/cosx}{2x} \bigg]$

$\sf \implies e^{\displaystyle \lim_{x \to 0} \bigg[\dfrac{-sin x}{2x (cos x)} \bigg]$

Now again it is in form of 0/0 indeterminant again apply L-HOSPITAL'S rule, we get.

$\sf \implies e^{\displaystyle \lim_{x \to 0} \bigg[ \dfrac{- cosx}{2(-x sin x + cos x)} \bigg]$

Put the value of x = 0 in equation, we get.

$\sf \implies e^{\bigg[\dfrac{-cos(0)}{2(-x sin(0) + cos(0)} \bigg]$

$\sf \implies e^{-1/2} = \dfrac{1}{\sqrt{e} }$

$\sf \implies \displaystyle \lim_{x \to 0} (cos x)^{1/x^{2} } = \dfrac{1}{\sqrt{e} }$

                                                                                                                         

MORE INFORMATION.

Some limits which do not exists.

$\sf (1)= \displaystyle \lim_{x \to 0} \bigg(\dfrac{1}{x} \bigg)$

$\sf (2)= \displaystyle \lim_{x \to 0} (x)^{1/x}$

$\sf (3)= \displaystyle \lim_{x \to 0} \dfrac{|x|}{x}$

$\sf (4)= \displaystyle \lim_{x \to a} \dfrac{|x - a|}{x - a}$

$\sf (5)= \displaystyle \lim_{x \to 0} sin \bigg(\dfrac{1}{x} \bigg)$

$\sf (6)= \displaystyle \lim_{x \to 0} cos\bigg(\dfrac{1}{x} \bigg)$

$\sf (7)= \displaystyle \lim_{x \to 0} (e)^{1/x}$

$\sf (8)= \displaystyle \lim_{x \to \infty} sin(x)$

$\sf (9)= \displaystyle \lim_{x \to \infty} cos(x)$