Prove $\lim _{x\to b^-}f^\ {'}\left(x\right)=\infty $ 
Let $f(x)$ be a differentiable function in $[a,b]$.
  Suppose $\lim _{x\to b^-}f\left(x\right)=\infty $ and $f^\ {'}(x)$ is monotonic, prove $\lim _{x\to b^-}f^\ {'}\left(x\right)=\infty $.
  Is the conclusion still true when $b=\infty?$

My try:
Let $x\in(a,b)$, so by MVT we get there exists $c\in(a,x)$ such that
$$f\:'\left(c\right)=\frac{f\left(x\right)-f\left(a\right)}{x-a}$$
The derivative must be monotonic increasing, otherwise we will get a contradicition to $\lim _{x\to b^-}f\left(x\right)=\infty $.
So we have $f^{\ '}(x)>f^{\ '}(c)$, now taking limits on both sides:
$$\lim _{x\to b^-}f^{\ '}(x)\ge \lim _{x\to b^-}\frac{f\left(x\right)-f\left(a\right)}{x-a}=\infty $$And we get the result. I think it is not true anymore when $b=\infty$, but not quite sure how to prove it.
Is my proof correct?
 A: The proof is correct if you mean differentiable in $[a,b)$ (obviously $f$ can't be continuous in $b$ if it diverges to $\infty$ there), as is the hunch for $b=\infty$ (not terribly formal, but we get the idea). Just consider $f(x)=\sqrt{x}$ in $[1,\infty)$ for example.
A: I think you need to change the question a bit.

Let $f$ be differentiable in $(a, b)$ and $\lim_{x \to b^{-}}f(x) = \infty$. If $f'$ is monotone on $(a, b)$ show that $\lim_{x \to b^{-}}f'(x) = \infty$. Does the same conclusion hold if $b = \infty$?

Since $f'$ is monotone, either $f'$ tends to a limit as $x \to b^{-}$ or tends to $\pm\infty$. If it tends to $-\infty$ then we can see that as $x \to b^{-}$ the derivative $f'$ is negative and hence $f$ is decreasing and hence there is no way it can tend to $\infty$. Thus we have either $f'(x) \to L$ or $f'(x) \to \infty$. Let us then assume that $f'(x) \to L$ as $x \to b^{-}$. Then we have number $c \in (a, b)$ such that $f'(x)$ is bounded in $(c, b)$. And then by mean value theorem we can see that $$f(x) = f(c) + (x - c)f'(d)$$ for $x \in (c, b)$ and some $d \in (c, x)$. Letting $x \to b^{-}$ we get an obvious contradiction as LHS tends to $\infty$ and the RHS is bounded. Thus the only option left is that $f'(x) \to \infty$.
When $b = \infty$ then you have many examples to convince that $f(x) \to \infty$ does not imply $f'(x) \to \infty$ even if $f'$ is monotone. One such example is $f(x) = \log x$.
