Question

If  \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\]  exists finitely, write the value of  \[\lim_{x \to c} f\left( x \right)\]

Answer 1

 Given:   

\[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]

\[\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)\]

Now,

\[\lim_{x \to c} f(x) = \lim_{x \to c} \left[ \left\{ \frac{f(x) - f(c)}{x - c} \right\} (x - c) + f(c) \right]\]
\[ = \lim_{x \to c} \left[ \left\{ \frac{f(x) - f(c)}{x - c} \right\} (x - c) \right] + f(c)\]
\[ = \lim_{x \to c} \left\{ \frac{f(x) - f(c)}{x - c} \right\} \lim_{x \to c} (x - c) + f(c)\]
\[ = f'(c) \times 0 + f(c)\]
\[ = f(c)\]