Question

If $\underset{x\to \infty }{\lim }x\sin \left(\frac{1}{x}\right)=A$ and $\underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)=B$, then which one of the following is correct?

• A. A = 1 and B = 0
• B. A = 0 and B = 1
• C. A = 0 and B = 0
• D. A = 1 and B = 1

Answer 1

Answer: (A)

As given $A=\underset{x\to \infty }{\lim }x\sin \left(\frac{1}{x}\right)$
$=\underset{x\to \infty }{\lim }\frac{\sin \left(\frac{1}{x}\right)}{\left(\frac{1}{x}\right)}$
Let $t=\frac{1}{x};x\to \alpha ,t\to 0$
$\Rightarrow A=\underset{t\to \infty }{\lim }\frac{\sin t}{t}=1$
$\left[\because \underset{t\to 0}{\lim }\frac{\sin x}{x}=1\right]$
and $B=\underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)$
$\Rightarrow B=0$
Therefore, $A=1$ and $B=0$ is correct