Q73 / 180 · KCET 2024 · Mathematics

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \cos x-1}{\cot x-1}$ is equal to

  • A
    $2$
  • B
    $\sqrt{2}$
  • Correct answer
    C
    $1 / 2$
  • D
    $1 / \sqrt{2}$

Solution

$\because \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \cos x-1}{\cot x-1}$ $=\lim _{x \rightarrow \frac{\pi}{4}} \frac{-\sqrt{2} \sin x}{-\operatorname{cosec}^2 x}$ [using L-Hospital rule] $=\frac{-\sqrt{2} \times \frac{1}{\sqrt{2}}}{-2}=\frac{1}{2}$

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

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