Q71 / 125 · BITSAT 2015 · Mathematics
If $\lim _{x \rightarrow} x \sin \left(\frac{1}{x}\right)=A$ and $\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=B$,
then which one of the following is correct?
- Correct answerA$A=1$ and $B=0$
- B$\quad A=0$ and $B=1$
- C$A=0$ and $B=0$
- D$A=1$ and $B=1$
Solution
As given,
$$
\mathrm{A}=\lim _{\mathrm{x} \rightarrow \infty} \mathrm{x} \sin \left(\frac{1}{\mathrm{x}}\right)=\lim _{\mathrm{x} \rightarrow \infty} \frac{\sin \left(\frac{1}{\mathrm{x}}\right)}{\left(\frac{1}{\mathrm{x}}\right)}
$$
Let $\mathrm{t}=\frac{1}{\mathrm{x}}$ when $\mathrm{x} \rightarrow \alpha, \mathrm{t} \rightarrow 0$
$$
\begin{aligned}
\Rightarrow \mathrm{A} &=\lim _{t \rightarrow \infty} \frac{\sin \mathrm{t}}{\mathrm{t}}=1 \\
&\left[\because \lim _{\mathrm{t} \rightarrow 0} \frac{\sin \mathrm{x}}{\mathrm{x}}=1\right]
\end{aligned}
$$
and $\mathrm{B}=\lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \sin \left(\frac{1}{\mathrm{x}}\right)$
$$
\Rightarrow \mathrm{B}=\lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \cdot \lim _{\mathrm{x} \rightarrow 0} \sin \left(\frac{1}{\mathrm{x}}\right)
$$
$$
\Rightarrow \mathrm{B}=0
$$
$\therefore \quad A=1$ and $B=0$ is correct
Hence, the correct answer is option: 'A'.