Calculate the limit $\lim\limits_{n\to\infty}\frac{\sqrt[n]{|x-1|}+1}{(x+1)^n+1}$ How can I compute the following limit? 

$$\lim_{n\to\infty}\frac{\sqrt[n]{|x-1|}+1}{(x+1)^n+1}$$
You have to look at different values of $x$. Look at $|x+1|&lt;1$ and $|x+1|\geq 1$ separately. In each case, what can you say about the numerator? Does it go to a finite limit? What can you say about the denominator? Does it blow up? Stay constant? Tend to a finite limit?

You'll have to use that $$|r|&lt;1\implies r^n\to 0$$ $$|r|&gt; 1\implies |r^n|\to  \infty$$ $$r=1\implies r^n\to 1$$  $$\forall r&gt;0\; ; \; r^{1/n}\to 1$$
$$\lim_{n\to\infty}\left({\sqrt[n]{|x-1|}+1\over (x+1)^n+1}\right)\to{\lim_{n\to\infty}\left({\sqrt[n]{|x-1|}+1}\right)\over\lim_{n\to\infty}\left({(x+1)^n+1}\right)}$$
$\lim_{n\to\infty}\left({(x+1)^n+1}\right)$ is only defined if x>-1, where it is infinity, so the limit is 0 becuase any number over infinity is 0;