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?

You may proceed as follows:


*

*Set $y = x-6$
$$\left(\frac{x+2}{x-6} \right)^{\frac{x}{4}} = \left(1 +\frac{8}{y} \right)^{\frac{y+6}{4}} = \left(1 +\frac{2}{\frac{y}{4}} \right)^{\frac{y}{4}}\cdot \left(1 +\frac{8}{y} \right)^{\frac{3}{2}} \stackrel{y \to \infty}{\longrightarrow}e^2$$
