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