Question

A function $- \frac{1}{2}$ is

• A. Not continuous at $\frac{1}{2}$
• B. Differentiable at $\frac{- e^{x} + 2^{x}}{x}$
• C. Continuous but not differentiable at $\left( \frac{2}{e} \right)$
• D. None of the above

Answer 1

Answer: (C)

Continuous but not differentiable at $\left( \frac{2}{e} \right)$

Answer 2

$\lim _ { h \rightarrow 0 ^ { - } } 1 + ( 2 - h ) = 3$, $\lim _ { h \rightarrow 0 ^ { + } } 5 - ( 2 + h ) = 3$, $f ( 2 ) = 3$

Hence, f is continuous at $x = 2$

Now $R f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { 5 - ( 2 + h ) - 3 } { h } = - 1$

$L f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { 1 + ( 2 - h ) - 3 } { - h } = 1$

$\because R f ^ { \prime } ( x ) \neq L f ^ { \prime } ( x )$

∴ f is not differentiable at $x = 2$.