Question

If f (x) is differentiable at x = c, then write the value of 

\[\lim_{x \to c} f \left( x \right)\]

Answer 1

Given: 

\[f(x)\]  is differentiable at 

\[x = c\]. Then,

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

or,   

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

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