Q211 / 270 · NDA 2013 (Phase 2) · Mathematics

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

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

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{1-\cos x}{x} \times \frac{(1+\cos x)}{(1+\cos x)} \\ & =\lim _{x \rightarrow 0} \frac{1-\cos ^2 x}{x(1+\cos x)}=\lim _{x \rightarrow 0} \frac{\sin ^2 x}{x \cdot(1+\cos x)} \\ & =\lim _{x \rightarrow 0} \frac{x}{(1+\cos x)} \cdot \frac{\sin ^2(x)}{x^2} \\ & =\frac{0}{(1+1)} \cdot 1=0\end{aligned}$

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

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