$\lim_{y\to\infty}\int_{0}^{\infty} (y\cos^2(x/y))/(y+x^4) \, dx$

Question

How do I calculate the limit

$$ \lim_{y\to\infty}\int_{0}^{\infty} \frac{y\cos^2(x/y)}{y+x^4} \, dx? $$

It's about measure theory.

I though about Fatou's lemma, but I couldn't solve it.

Answer 1

The change of variable $x=y^{1/4}z$ shows that the integral is $$I(y)=\int_{0}^{\infty} \frac{y\cos^2(x/y)}{y+x^4} \, dx=y^{1/4}\int_{0}^{\infty} \frac{\cos^2(z/y^{3/4})}{1+z^4} \, dz. $$ The integral on the RHS converges to $$\int_{0}^{\infty} \frac1{1+z^4} \, dz=c\gt0, $$ hence $I(y)\sim cy^{1/4}$, and the limit follows.

Answer 2

Let $f(x,y)=\frac{y\cos^2(x/y)}{y+x^4}$; this is a positive function. We have $\lim_{y\to\infty}f(x,y)=1$ for every $x>0$. By Fatou's lemma $$ \int_0^\infty\liminf_{y\to\infty}f(x,y) dx \leq \liminf_{y\to\infty}\int_0^\infty f(x,y) dx. $$ The LHS is $\int_0^\infty1dx=\infty$, so the desired limit is infinite.