Q90 / 160 · WBJEE 2010 · Mathematics

The value of $\operatorname{Lt}_{x \rightarrow 0} \frac{\sin ^2 x+\cos x-1}{x^2}$ is

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

Solution

Hints: $\operatorname{Lim}_{x \rightarrow 0} \frac{\sin ^2 x+\cos x-1}{x^2}=\operatorname{Lim}_{x \rightarrow 0} \frac{\cos x-\cos ^2 x}{x^2}=\operatorname{Lim}_{x \rightarrow 0}\left(\frac{1-\cos x}{x^2}\right) \cos x$ $$ =\operatorname{Lim}_{x \rightarrow 0} \frac{2 \sin ^2 \frac{x}{2}}{\left(\frac{x}{2}\right)^2 \times 4}=\frac{1}{2} $$

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

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