real analysis - Limit of $limlimits_{n rightarrow infty}(sqrt{x^8+4}-x^4)$

I have to determine the following:

$\lim\limits_{n \rightarrow \infty}(\sqrt{x^8+4}-x^4)$

$\lim\limits_{n \rightarrow \infty}(\sqrt{x^8+4}-x^4)=\lim\limits_{x \rightarrow \infty}(\sqrt{x^8(1+\frac{4}{x^8})}-x^4 = \lim\limits_{x \rightarrow \infty}(x^4\sqrt{1+\frac{4}{x^8}}-x^4 = \lim\limits_{x \rightarrow \infty}(x^4(\sqrt{1+\frac{4}{x^8}}-1)= \infty$

Could somebody please check, if my solution is correct?