Limits of difference quotient

Question

Suppose $f(x)$ is continuous, then $\lim_{h \to 0} f(x+h) = f(x)$ then what is the $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$?

Does this limit always exist?

can anyone provides me a continuous f(x) where this limit exists and another continuous f(x) where it does not exist?

Then you're basically asking for examples of a differentiable function and one which is not differentiable. — Keen-ameteur, Sep 21 '20 at 07:12
If you had made some attempt you would have come up with an example for the first case at least. (How about $f\equiv 0$?) I am voting this question for closure. — Kavi Rama Murthy, Sep 21 '20 at 07:36
It seems a bit odd that this guy has posted questions about muntivariable integral calculus yet apparently is confused by this fairly simple high school calculus problem? Seems like this might be a troll question. — Snacc, Sep 21 '20 at 20:31

score 0 · Accepted Answer · answered Sep 21 '20 at 07:11

0

Consider both $f(x)=x$ and $f(x)=|x|$ at $x=0$. Check continuity and compute the above limit.

answered Sep 21 '20 at 07:11

1 Answers1