Question

The function $f(x)$ at $\infty$

• A. Is continuous but not differentiable
• B. Is discontinuous
• C. Is having continuous derivative
• D. Is continuous and differentiable

Answer 1

Answer: (D)

Is continuous and differentiable

Answer 2

$\lim _ { x \rightarrow 0 } f ( x ) = x ^ { 2 } \sin \left( \frac { 1 } { x } \right)$ but $- 1 \leq \sin \left( \frac { 1 } { x } \right) \leq 1$ and $x \rightarrow 0$

∴ $\lim _ { x \rightarrow 0 ^ { + } } f ( x ) = 0 = \lim _ { x \rightarrow 0 ^ { - } } f ( x ) = f ( 0 )$

Therefore $f ( x )$ is continuous at $x = 0$. Also, the function $f ( x ) = x ^ { 2 } \sin \frac { 1 } { x }$ is differentiable because

$R f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { h ^ { 2 } \sin \frac { 1 } { h } - 0 } { h } = 0$,$L f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { h ^ { 2 } \sin \left( \frac { 1 } { - h } \right) } { - h } = 0$.