Q225 / 270 · NDA 2017 (Phase 2) · Mathematics

If $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x}{x}=l$ and $\lim _{x \rightarrow \infty} \frac{\cos x}{x}=m$, then which one of the following is correct?

  • A
    $l=1, \mathrm{~m}=1$
  • B
    $l=\frac{2}{\pi}, \mathrm{~m}=\infty$
  • Correct answer
    C
    $l=\frac{2}{\pi}, \mathrm{~m}=0$
  • D
    $l=1, \mathrm{~m}=\infty$

Solution

$\begin{aligned} & \lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x}{x}=l \text { and } \lim _{x \rightarrow \infty} \frac{\cos x}{x}=m \\ & \lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x}{x}=\frac{1}{\frac{\pi}{2}}=\frac{2}{\pi} ; \lim _{x \rightarrow \infty} \frac{\cos x}{x}=0\end{aligned}$ $\therefore l=\frac{2}{\pi}, \mathrm{~m}=0$

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

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