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BITSAT 2022
Exam
BITSAT
Year
2022
Subject
Mathematics
Chapter
Limits

BITSAT 2022 · Mathematics · Limits

The value of _ x arrow 0 (1- (1- x))/(x^4) is

The value of lim⁡x→01−cos⁡(1−cos⁡x)x4\lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^4} is

Options

  1. A16\frac{1}{6}
  2. B18\frac{1}{8}
  3. C110\frac{1}{10}
  4. D112\frac{1}{12}

Correct answer

B. 18\frac{1}{8}

Step-by-step solution

1−cos⁡(1−cos⁡x)=2sin⁡2(1−cos⁡x2)=2sin⁡2(sin⁡2x2)∴lim⁡x→01−cos⁡(1−cos⁡x)x4=lim⁡x→02sin⁡2(sin⁡2x2)x4=lim⁡x→02sin⁡2(sin⁡2x2)(sin⁡2x2)2×sin⁡4x2(x2)4×16=181-\cos (1-\cos x)=2 \sin ^2\left(\frac{1-\cos x}{2}\right)=2 \sin ^2\left(\sin ^2 \frac{x}{2}\right)\therefore \lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^4}=\lim _{x \rightarrow 0} \frac{2 \sin ^2\left(\sin ^2 \frac{x}{2}\right)}{x^4}=\lim _{x \rightarrow 0} \frac{2 \sin ^2\left(\sin ^2 \frac{x}{2}\right)}{\left(\sin ^2 \frac{x}{2}\right)^2} \times \frac{\sin ^4 \frac{x}{2}}{\left(\frac{x}{2}\right)^4 \times 16}=\frac{1}{8}
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