How to prove of $\lim \frac{1}{|x|+1}=0$ as $x$ goes to $-\infty$?

Question

I understand that $\displaystyle \lim_{x\rightarrow \infty} \frac{1}{|x|+1}=0$.

Let $\varepsilon>0 $ and I choose $M>0$ such that $\frac{1}{M+1}<\varepsilon$. Thus, if $x\geq M$ then $$\left|\frac{1}{|x|+1}\right|= \frac{1}{|x|+1}=\frac{1}{x+1}\leq \frac{1}{M+1}<\varepsilon.$$

But I cannot figure out how to prove that $\displaystyle \lim_{x\rightarrow -\infty} \frac{1}{|x|+1}=0$.

How can you prove this case? Any hints or tips will be appreciated. Thanks in advanced.

Answer 1

If $x <-M$ then $|x| >M$ so $\frac 1 {|x|+1} <\frac 1 {M+1} <\epsilon$.