Let $f:(0,\infty)\rightarrow\mathbb{R}$ be differentiable and let $b$ be a real number. Prove that if $f'(t)\rightarrow b$ as $t\rightarrow\infty$, then $\frac{f(t)}{t}\rightarrow b$ as $t\rightarrow\infty$.
Median Value Theorem: If $f$ is continuous and differentiable, then $\exists c$ such that $a<c<b$ and $f'(c)=\frac{f(b)-f(b)}{b-a}$. So when $c\rightarrow\infty$, $a,b\rightarrow\infty$. How do I get $\frac{f(t)}{t}$?
