Improper integral calculation - limit at infinity Will you please help me prove the following limit is zero ? 


  $$\lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \cos(x u) du.  
$$


Thanks in advance
If you don't want to appeal to Riemann-Lebesgue, you could also try integrating by parts (being careful to justify the differentiability at 0 of $u\mapsto (1-{\rm e}^{-u^4})/u^2$).
Hint: Check Riemann-Lebesgue lemma