How to find the limit: $$\lim_{x \rightarrow \infty} \frac{|x|^{n} - (1 + |x|^2)^{\frac{n}{2}}}{(1 + |x|^2)^{\frac{n}{2}}}.$$ where $1<n<2$ and $x \in \mathbb R^d$. It seems the limit is 0 by using rationalization (conjugation), but how to deal with the numerator after multiplying the conjugate term?
Update on the range of $n$: Thanks for the comment below and I'm sorry for the confusion.
Let $u=1+|x|^2$, so that $|x|^n=(u-1)^{n/2}$. We have
$${|x|^n-(1+|x|^2)^{n/2}\over(1+|x|^2)^{n/2}}={(u-1)^{n/2}-u^{n/2}\over u^{n/2}}=\left(1-{1\over u}\right)^{n/2}-1\to1-1=0$$
since $u\to\infty$ as $|x|\to\infty$.