Question:
If $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely, write the value of $\lim _{x \rightarrow c} f(x)$.
Solution:
Given: $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. Then,
$\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$
Now,
$\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$
$=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)\right]+f(c)$
$=\lim _{x \rightarrow c}\left\{\frac{f(x)-f(c)}{x-c}\right\} \lim _{x \rightarrow c}(x-c)+f(c)$
$=f^{\prime}(c) \times 0+f(c)$
$=f(c)$