Proof of $\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$ Is there an easy way to prove that $$\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$$ with $x\in \mathbb R$ without using $\epsilon-\delta$ definition?
Any help would be really appreciated 

If $|x|<1$, then $|x^k-x|=x-x^k$
$$\lim_{k\rightarrow\infty}\frac{x-x^k}{1+x-x^k}=\frac{x}{1+x}.$$
If $|x|=1$, then
$$\lim_{k\rightarrow\infty}\frac{|x^k-x|}{1+|x^k-x|}=0.$$
If $|x|>1$, then $|x^k-x|=x^k-x$
$$\lim_{k\rightarrow\infty}\frac{x^k-x}{1+x^k-x}=1-\lim_{k\rightarrow\infty}\frac{1}{1+x^k-x}=1.$$
