Question

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

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