Q234 / 270 · NDA 2022 (Phase 2) · Mathematics

What is $\lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\cos ^4 x}}$ equal to?

  • A
    $\frac{1}{2 \sqrt{2}}$
  • B
    $-\frac{1}{2 \sqrt{2}}$
  • C
    $\sqrt{2}$
  • Correct answer
    D
    Limit does not exist

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\cos 4 x}}=\lim _{x \rightarrow 0} \frac{2 x}{\sqrt{2}|\sin 2 x|} \times \frac{1}{2} \\ & \text { L.H.L }=-\frac{1}{2 \sqrt{2}} \text { and R.H.L. }=\frac{1}{2 \sqrt{2}} \\ & \because \quad \text { L.H.L } \neq \text { R.H.L. } \end{aligned}$ So, $\lim _{x \rightarrow 0} f(x)$ is not exist.

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

Relevant Questions