Solve $\displaystyle\lim_{x \to \frac{\pi}{2}}{\biggl[(1+\cot x)^{\tan x}\biggr]^3}$

Question

Evaluate the limit: $$\displaystyle\lim_{x \to \frac{\pi}{2}}{\biggl[(1+\cot x)^{\tan x}\biggr]^3}$$

Could someone help me to solve this limit?

I found the answer in symbolab, however I could not understand how the steps are explained.

Please explain this to me. Thank you.

Answer 1

Let $y = \tan x$. Then:

$$\displaystyle\lim_{x \to \frac{\pi}{2}}{\biggl[(1+\cot x)^{\tan x}\biggr]^3} = \displaystyle\lim_{y \to \infty}{\biggl[\left(1+\frac{1}{y}\right)^{y}\biggr]^3} = e^3.$$

Answer 2

Although the easier way has already been shown, we usually solve the limits of the form $1^{\infty}$ by taking the natural logarithm on both sides.

So, if we call the limit as $L$, then we have

$\begin{align} \ln L&=\displaystyle \lim_{x \to \frac{\pi}{2}}\ln {\biggl[(1+\cot x)^{\tan x}\biggr]^3}\\ &=\lim_{x \to \frac{\pi}{2}} 3\tan x \ln {(1+\cot x)} \\ &=\lim_{x \to \frac{\pi}{2}} \frac{3\ln {(1+\cot x)}}{\cot x} \\ &=3 \hspace{100pt} \left(\text{Since} \color{red}{\lim_{x \to 0}\frac{\ln (1+x)}{x} =1}\right) \\ \end{align}$

Therefore $L=e^3$.