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%5E%7B%5Cfrac%7Bx%7D%7B4%7D%7D) calculator solves it is not immediately clear to me, is there any other way to find the above limit?

**HINT**

We have that

$$\left(\frac{x+2}{x-6}\right)^{\frac x 4}=\left(\frac{x-6+8}{x-6}\right)^{\frac x 4}=\left(1+\frac{8}{x-6}\right)^{\frac x 4}$$

then we can manipulate further in order to use the standard limit.

Refer to the related

* Calculating a limit with trignonometeric and quadratic function