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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Limits

NDA 2019 · Mathematics · Limits

_ x arrow ∞ (1- ^3 4 x)/(x^2) is equal to

lim⁡x→∞1−cos⁡34xx2\lim _{x \rightarrow \infty} \frac{1-\cos ^3 4 x}{x^2} is equal to

Options

  1. A0
  2. B12
  3. C24
  4. D36

Correct answer

C. 24

Step-by-step solution

lim⁡x→01−cos⁡34xx2\lim _{x \rightarrow 0} \frac{1-\cos ^3 4 x}{x^2} The given limit is in 00\frac{0}{0} form. So, apply L'hospital rule. lim⁡x→0−3cos⁡24x(−sin⁡4x)42x=lim⁡x→012cos⁡24xsin⁡4x(2x)2×2=24⋅lim⁡x→0cos⁡24x⋅sin⁡4x4x=24(1)=24\begin{aligned} & \lim _{x \rightarrow 0} \frac{-3 \cos ^2 4 x(-\sin 4 x) 4}{2 x} \\ & =\lim _{x \rightarrow 0} \frac{12 \cos ^2 4 x \sin 4 x}{(2 x) 2} \times 2 \\ & =24 \cdot \lim _{x \rightarrow 0} \cos ^2 4 x \cdot \frac{\sin 4 x}{4 x} \\ & =24(1)=24\end{aligned}
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