Limits: Can't understand this worked example

Question

I can't seem to understand the following given example while working with Limits.

$$\lim\limits_{x \to \infty}({x\over1+x})^x = \lim\limits_{x\to \infty}({x +1 -1\over1+x})^x = \lim\limits_{x \to \infty}(1- {1\over1+x})^x = \lim\limits_{x \to \infty}(1+ {1\over-1-x})^{(-1-x)({x\over-1-x})} = e^{-1}$$

I understand most of it, except for this part; $$\lim\limits_{x \to \infty}({x +1 -1\over1+x})^x = \lim\limits_{x \to \infty}(1- {1\over1+x})^x$$

What happends to the x in the numerator? It seems to have been left out but I wouldn't see how this could have been done in a correct / proper way.

If anyone could give and hints or suggestions, that'd be great.

Thanks in advance, Michiel

Answer 1

$\frac{x+1-1}{x+1}=\frac{x+1}{x+1}-\frac{1}{x+1}$

Answer 2

Just in case.... $$\lim\limits_{x \to \infty}\left({x +1 -1\over1+x}\right)^x = \lim\limits_{x \to \infty}\left({x +1\over 1+x} -{1\over1+x}\right)^x = \lim\limits_{x \to \infty}\left(1- {1\over1+x}\right)^x $$