Question

evaluate lim(cos x )^cot^2x​

Answer 1

EXPLANATION.

$\sf \implies \lim_{x \to 0} (cosx)^{cot^{2} x}$

As we know that,

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

$\sf \implies \lim_{x \to 0} (cos(0))^{cot^{2}(0) }$

$\sf \implies \lim_{x \to 0} 1^{\infty}$

We can see that it is a form of 1^∞,

So, we can take log on both sides, we get.

$\sf \implies x = \lim_{x \to 0} (cosx)^{cot^{2} x}$

$\sf \implies log(x) = \lim_{x \to 0} log \bigg(cosx\bigg)^{cot^{2} x}$

$\sf \implies log(x) = \lim_{x \to 0} (cot^{2}x )log (cosx)$

$\sf \implies log(x) = \lim_{x \to 0} \dfrac{log(cosx)}{tan^{2} x}$

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

$\sf \implies log(x) = \lim_{x \to 0} \dfrac{log(cos(0))}{tan^{2}(0) }$

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

Now we can apply L-Hopital rule, we get.

$\sf \implies log(x) = \lim_{x \to 0} log \bigg[\dfrac{1 \times (-sin x)}{cos x} \times \dfrac{1}{(tan x)^{2} }\bigg]$

Differentiate (tan x)².

⇒ d(tan x)²/dx = 2.tan x. sec²x.

$\sf \implies log(x) = log \bigg[\dfrac{-sin x}{cos x} \times \dfrac{1}{2.tanx.sec^{2}x }\bigg]$

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

$\sf \implies log(x) = log \bigg[\dfrac{-sin(0)}{cos(0)} \times \dfrac{1}{2.tan(0).sec^{2}(0) } \bigg]$

$\sf \implies log(x) = log(0)$

$\sf \implies log(x) = 1.$

                                                                                       

MORE INFORMATION.

Some limits which do not exists.

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

$\sf \implies (2) = \lim_{x \to 0} \bigg((x)^{\dfrac{1}{x} } \bigg)$

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

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

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

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

$\sf \implies (7) = \lim_{x \to 0} \bigg( (e)^{\dfrac{1}{x} } \bigg)$

$\sf \implies (8) = \lim_{x \to \infty} Sin(x)$

$\sf \implies (9) = \lim_{n \to \infty} Cos(x)$

Answer 2

Answer:

PLEASE SEE THE ABOVE ATTACHMENT