Question

If ${lim}_{x\to \infty }x\sin \left(\frac{1}{x}\right)=A$ and ${lim}_{x\to 0}x\sin \left(\frac{1}{x}\right)=B$, 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)

$A=1$ and $B=0$

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