$\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a} = \lim_{x\rightarrow a} f'(c_x) = \lim_{c_x\rightarrow a} f'(c_x)$
Where $f:\mathbb{R}\rightarrow \mathbb{R}$, and $c_x$ is some value between $x$ and $a$ by the mean value theorem.
- How do you rigorously show that as $x\rightarrow a$, then $c_x\rightarrow a$ because it is always between $x$ and $a$? Something like $|x-a|<\delta \Rightarrow |c_x-a|<\delta$? I feel that just switching $c_x$ for $x$ above is kind of non-rigorous (or completely incorrect).
- I think my use of $c_x$ is nonstandard; I am trying to express dependence on $x$. Is there a better way to write what I have written above?
Thank you in advance!