How to transform this limit expression as a limit of $e$ I have the following expression:
$\displaystyle\lim _{x\to \infty }\left(\dfrac{x+2}{\:x-6}\right)^\left(\dfrac{x}{\:4}\right)$
I'm studying Calculus I and our lector has shown us ways of transforming such limits to:
$\displaystyle\lim_{x\to \infty }\left(1+\frac{1}{\:x}\right)^x = e$
The way this calculator solves it is not immediately clear to me, is there any other way to find the above limit?

$\displaystyle\lim _{x\to \infty }\left(\dfrac{x+2}{\:x-6}\right)^\left(\dfrac{x}{\:4}\right)=\lim _{x\to \infty }\left(\dfrac{1+\dfrac2x}{1-\dfrac6x}\right)^\left(\dfrac{x}{\:4}\right)=\dfrac{\left(\lim_{x\to\infty}\left(1+\dfrac2x\right)^{x/2}\right)^{1/2}}{\left(\lim_{x\to\infty}\left(1-\dfrac6x\right)^{-x/6}\right)^{-3/2}}=\dfrac{e^{1/2}}{e^{-3/2}}=?$
