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Question: Answered & Verified by Expert
$\lim _{x \rightarrow 0}\left(\frac{a^x-b^x}{x}\right)=$
MathematicsLimitsJEE Main
Options:
  • A $\log \left(\frac{b}{a}\right)$
  • B $\log \left(\frac{a}{b}\right)$
  • C $\frac{a}{b}$
  • D $\log a^b$
Solution:
1223 Upvotes Verified Answer
The correct answer is: $\log \left(\frac{a}{b}\right)$
$\begin{aligned} \lim _{x \rightarrow 0} \frac{a^x-b^x}{x} & =\lim _{x \rightarrow 0}\left(\frac{a^x-1}{x}\right)-\lim _{x \rightarrow 0}\left(\frac{b^x-1}{x}\right) \\ & =\log a-\log b=\log (a / b) .\end{aligned}$

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