Q114 / 180 · COMEDK 2019 · Mathematics

$\lim _{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)$ is equal to

  • A
    $\infty$
  • B
    0
  • Correct answer
    C
    2
  • D
    $\frac{1}{2}$

Solution

We have, $$ \begin{aligned} &\lim _{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)=\lim _{x \rightarrow \infty} \frac{\sin \left(\frac{2}{x}\right)}{\left(\frac{2}{x}\right)} \times 2 \\ &\text { Put } \frac{2}{x}=y . \\ &\text { Then } x \rightarrow \infty \\ &\Rightarrow \frac{2}{x} \rightarrow 0 \Rightarrow y \rightarrow 0 \\ &\therefore \quad \lim _{x \rightarrow \infty} \frac{\sin \left(\frac{2}{x}\right)}{\left(\frac{2}{x}\right)} \times 2 \\ &\quad=\lim _{y \rightarrow 0} \frac{\sin y}{y} \times 2=1 \times 2=2 \end{aligned} $$

Hence, the correct answer is option: 'C'.

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