Limit with denominator = $0$

Question

Prove that :

$$\lim_{x\to t} \frac{f(x)-m}{x-t} = k$$

if $k \in C ( or R)$ than $f(t) = m $

Answer 1

Note that if

$$f(x)\to n\neq m \implies \lim_{x\to t} \frac{f(x)-m}{x-t} = \pm \infty $$

thus we need that $f(x)\to m$ but note that this is a necessary but not a sufficient condition.

Indeed for example assuming

$$f(x)=m+|x-t|\implies \lim_{x\to t} \frac{f(x)-m}{x-t}=\lim_{x\to t} \frac{|x-t|}{x-t}$$

which doesn't exist.