Originally Posted by

**CaptainBlack** If $\displaystyle b>0$ then for $\displaystyle x$ large enough $\displaystyle f'(x)>\epsilon>0$, and so for $\displaystyle x$ large enough and all $\displaystyle y>0$ (by a simple corollary to the mean value theorem):

$\displaystyle f(x+y)>f(x)+\epsilon y$

hence the limit of this as $\displaystyle y \to \infty$ cannot be finite, which contradicts the assumption that $\displaystyle \lim_{x \to \infty}f(x)$ is finite.

A similar argument applies when $\displaystyle b<0$.

Now you will need to fill in the detail to complete this.

CB