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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Limits

NDA 2013 · Mathematics · Limits

What is _ x arrow 0 (1- x)/(x) equal to?

What is lim⁡x→01−cos⁡xx\lim _{x \rightarrow 0} \frac{1-\cos x}{x} equal to?

Options

  1. A0
  2. B12\frac{1}{2}
  3. C1
  4. D2

Correct answer

A. 0

Step-by-step solution

lim⁡x→01−cos⁡xx×(1+cos⁡x)(1+cos⁡x)=lim⁡x→01−cos⁡2xx(1+cos⁡x)=lim⁡x→0sin⁡2xx⋅(1+cos⁡x)=lim⁡x→0x(1+cos⁡x)⋅sin⁡2(x)x2=0(1+1)⋅1=0\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}
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