Question

If $\underset{x\to c}{lim}\frac{f\left(x\right)-f\left(c\right)}{x-c}$ exists finitely, then

• A. $\underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$
• B. $\underset{x\to c}{lim}{f}^{\prime }\left(x\right)=0$
• C. $\underset{x\to c}{lim}f\left(x\right)$ does not exist
• D. $\underset{x\to c}{lim}f\left(x\right)$ may or may not exist

Answer 1

The correct option is A $\underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$

$\underset{x\to c}{lim}\frac{f\left(x\right)-f\left(c\right)}{x-c}=f^{\prime }\left(c\right)$
It is given that $f^{\prime }\left(c\right)$ exist $\Rightarrow f$ is differentiable at $x=c$
And we know that, if $f$ is differetiable at any point then it must be continuous at that point
$\therefore \underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$