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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Limits

NDA 2010 · Mathematics · Limits

What is _ x arrow 0 (a^x-b^x)/(x) equal to?

What is lim⁡x→0ax−bxx\lim _{x \rightarrow 0} \frac{a^x-b^x}{x} equal to?

Options

  1. Aln⁡(ab)\ln (a b)
  2. Bln⁡aln⁡b\frac{\ln a}{\ln b}
  3. Cln⁡(ab)\ln \left(\frac{a}{b}\right)
  4. Dln⁡(ba)\ln \left(\frac{b}{a}\right)

Correct answer

C. ln⁡(ab)\ln \left(\frac{a}{b}\right)

Step-by-step solution

lim⁡x→0ax−bxx=lim⁡x→0axlog⁡ea−bxlog⁡eb1\begin{aligned} & \lim _{x \rightarrow 0} \frac{a^x-b^x}{x} \\ & =\lim _{x \rightarrow 0} \frac{a^x \log e^a-b^x \log e^b}{1} \end{aligned}(usingL′Hospital′sRule)(using L'Hospital's Rule)=log⁡ea−log⁡eb=log⁡(ab)\begin{aligned} & =\log e^a-\log e^b \\ & =\log \left(\frac{a}{b}\right)\end{aligned}$
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