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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Limits

NDA 2021 · Mathematics · Limits

If _ x arrow a (a^x-x^a)/(x^a-a^a)=-1 then what is the value of a ?

If lim⁡x→aax−xaxa−aa=−1\lim _{x \rightarrow a} \frac{a^x-x^a}{x^a-a^a}=-1 then what is the value of aa ?

Options

  1. A−1-1
  2. B00
  3. C11
  4. D22

Correct answer

C. 11

Step-by-step solution

lim⁡x→aax−xaxa−aa\lim _{x \rightarrow a} \frac{a^x-x^a}{x^a-a^a} This is in 00\frac{0}{0} form, by L'Hospital rule lim⁡x→aax−xaxa−aa=lim⁡x→aax⋅ln⁡a−a⋅xa−1a⋅xa−1−0=aa⋅ln⁡a−aaaa=ln⁡a−1\lim _{x \rightarrow a} \frac{a^x-x^a}{x^a-a^a}=\lim _{x \rightarrow a} \frac{a^x \cdot \ln a-a \cdot x^{a-1}}{a \cdot x^{a-1}-0}=\frac{a^a \cdot \ln a-a^a}{a^a}=\ln a-1. ∴ln⁡a−1=−1⇒ln⁡a=0∴a=(e)0=1.\begin{aligned} & \therefore \ln a-1=-1 \Rightarrow \ln a=0 \\ & \therefore a=(e)^0=1 .\end{aligned}
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