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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,

$\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$

$=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)\right]+f(c)$

$=\lim _{x \rightarrow c}\left\{\frac{f(x)-f(c)}{x-c}\right\} \lim _{x \rightarrow c}(x-c)+f(c)$

$=f^{\prime}(c) \times 0+f(c)$

$=f(c)$

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