horizontal asymptotes and derivatives Suppose that $f(x)$ has a horizontal asymptote. Must it be the case that the derivative approaches $0$ as $x$ tends to infinity? I do not think so, and I think I have a counter example, but I have yet to prove it. 
Of course, I know that the converse is not true (a derivative approaching $0$ need not come from a function with a horizontal asymptote... think $\ln x, \sqrt x$, etc).

If $~f(x)$ is a function that is differentiable on the real line and $~y=a$ is a horizontal asymptote of $f(x)$, then can we say that $~y=0$ is a horizontal asymptote of $~f'(x)$? 
I think this is true. Any help is appreciated.
