How can I use epsilon-delta to solve this problem?

Question

1. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous and differentiable at every point $x\in\mathbb{R}−\{c\}$ for some $c\in\mathbb{R}$.

2. $\lim_{x\to c}f'(x)=C$.

How can I prove the sentence below? $$ \lim_{x→c}\frac{f(x)−f(c)}{x−c}=C $$ I know that I have to use mean-value theorem and epsilon-delta, but I don't know exactly how can I use the epsilon-delta to prove this problem.

Answer 1

Take a sequence $x_n\uparrow c.$ By the mean value theorem, for each $n$ there is a $x_n<d_n<c$ such that $$\frac{f(x_n)−f(c)}{x_n−c}=f'(d_n)$$ As $x_n\uparrow c$, $d_n\to c$ so we have $$\lim_{x\uparrow c}\frac{f(x)−f(c)}{x−c}=\lim_{x\uparrow c}f'(x)=C,$$ and similarly for $x_n\downarrow c.$