$L^p$ convergence of this sequence

Question

The sequence $$f_h(x)=\frac{h^2 x}{h^4+x^4}$$ for $x \in ]0,+\infty[$ converges pointwise to the function $f(x)=0$. I have to find out for which $p \in [1,+\infty]$ $f_h$ converges in $L^p(]0,+\infty[)$. Can you help me?

$$||f_h-f||_p=\int_0^{+\infty} \frac{h^{2p}x^p}{(h^4+x^4)^p}dx <+\infty.$$ Does $||f_h||_p \rightarrow 0$ as $h \rightarrow 0$?

Thanks.

Answer 1

Hint: There exists a number $\alpha=\alpha_p$ such that $$\int_0^{+\infty} \frac{h^{2p}x^p}{(h^4+x^4)^p}dx=h^\alpha\int_0^{+\infty} \frac{1}{(1+x^4)^p}dx.$$Figure out what $\alpha$ is and you're set.