Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly 
positive. If $x\_n$ is a sequence of points such that $f(x\_n)\rightarrow 0$, show that (or 
give a counterexample) the gradient $\nabla f(x\_n)$ also tends to zero. 



Asymptotic behavior of convex functions
A counterexample is
$$f=\sqrt{y^2+e^{-x}}.$$
You can verify by computing the second derivatives that this is convex.
As a sequence $x\_n$ you can take $(n,1/n)$. Then $f(x\_n)\to 0$ but the derivative with respect to $y$ tends to 1. Thus the gradient does not tend to 0.


