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BITSAT 2021
Exam
BITSAT
Year
2021
Subject
Mathematics
Chapter
Limits

BITSAT 2021 · Mathematics · Limits

Value of _ x arrow 0 a^ x -1 x is

Value of lim⁡x→0asin⁡x−1sin⁡x\lim _{x \rightarrow 0} \frac{a^{\sin x}-1}{\sin x} is

Options

  1. Alog⁡a\log \mathrm{a}
  2. Bsin⁡x\sin x
  3. Clog⁡(sin⁡x)\log (\sin x)
  4. Dcos⁡x\cos x

Correct answer

A. log⁡a\log \mathrm{a}

Step-by-step solution

We have, lim⁡x→0asin⁡x−1sin⁡x=lim⁡y→0ay−1y=log⁡a\lim _{x \rightarrow 0} \frac{a^{\sin x}-1}{\sin x}=\lim _{y \rightarrow 0} \frac{a^{y}-1}{y}=\log \mathrm{a}, where y=sin⁡x∵x→0⇒y=sin⁡x→0\mathrm{y}=\sin \mathrm{x}\because \mathrm{x} \rightarrow 0 \Rightarrow \mathrm{y}=\sin \mathrm{x} \rightarrow 0
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