Prove the limit problems I got two problems asking for the proof of the limit: 

Prove the following limit: $$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$

and, 

Prove the following limit: $$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \, dp=1.$$

I may feel that these two problems are of the same kind. World anyone please help me with one of them and I may figure out the other one? Many thanks! 

Let
$$ f(x)=\ x e^{x^2}\int_x^\infty e^{-t^2} \implies f(x)=\ x e^{x^2}g(x).$$  
We can see that $ f(0)=0 $ and $f(x)>0,\,\, \forall x>0$. Taking the limit as $x$ goes to infinity and using L'hobital's rule and Leibniz integral rule yields
$$ \lim_{ x\to \infty } xe^{x^2}g(x) = \lim _{x\to \infty} \frac{g(x)}{\frac{1}{xe^{x^2}}}=\lim_{x \to \infty} \frac{g'(x)}{\frac{1}{(xe^{x^2})'}}=\lim_{x \to \infty} \frac{-e^{-x^2}}{{-{\frac {{{\rm e}^{-{x}^{2}}} \left( 2\,{x}^{2}+1 \right) }{{x}^{2}}}}} =\frac{1}{2}. $$  
