Q226 / 270 · NDA 2021 (Phase 1) · Mathematics

If $\lim _{x \rightarrow a} \frac{a^x-x^a}{x^a-a^a}=-1$ then what is the value of $a$ ?

  • A
    $-1$
  • B
    $0$
  • Correct answer
    C
    $1$
  • D
    $2$

Solution

$\lim _{x \rightarrow a} \frac{a^x-x^a}{x^a-a^a}$ This is in $\frac{0}{0}$ form, by L'Hospital rule $\lim _{x \rightarrow a} \frac{a^x-x^a}{x^a-a^a}=\lim _{x \rightarrow a} \frac{a^x \cdot \ln a-a \cdot x^{a-1}}{a \cdot x^{a-1}-0}$ $=\frac{a^a \cdot \ln a-a^a}{a^a}=\ln a-1$. $\begin{aligned} & \therefore \ln a-1=-1 \Rightarrow \ln a=0 \\ & \therefore a=(e)^0=1 .\end{aligned}$

Hence, the correct answer is option: 'C'.

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