Question

Assertion(A): $f\left(x\right)=\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$ is continuous at $x=0$
Reason(R): Both $h\left(x\right)=x^2,g\left(x\right)=\{\begin{array}{cc}\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$are continuous at $x=0$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. R is true but A is false

Answer 1

Answer: (C)

A is true but R is false

Assertion

$f\left(0\right)=0$

$\underset{x\to 0}{lim}{x}^2\sin \left(\frac{1}{x}\right)=0^2\times$ (finite value)

$=0$

$\therefore$ It is continuous at $x=0$

Reason: $h\left(x\right)=x^2$ is continuous

but $g\left(x\right)$ is not continuous

$\underset{x\to 0}{lim}\sin \left(\frac{1}{x}\right)=$not defined (value oscillates)

$\underset{x\to 0}{lim}\sin \left(\frac{1}{x}\right)\ne 0$

$\therefore$ not continuous.