Question

If $\underset{x\to a}{\lim }\frac{a^x-x^a}{x^x-a^a}=-1$ then $a$ is equal to

• A. 0
• B. e
• C. 2
• D. None of the above

Answer 1

Answer: (D)

We have, $\underset{x\to a}{\lim }\frac{a^x-x^a}{x^x-a^a}=-1$
$\Rightarrow \underset{x\to a}{\lim }\frac{a^x\log a-a{x}^{a-1}}{x^x(\log x+1)}=-1$
[using L'Hospital's rule]
$\Rightarrow \frac{a^a\log a-a\cdot a^{a-1}}{a^a(\log a+1)}=-1$
$\Rightarrow \frac{\log a-1}{\log a+1}=-1$
$\Rightarrow \log a-1=-\log a-1$
$\Rightarrow 2\log a=0$
$\Rightarrow \log a=0$
$\Rightarrow a=e^0$
$\Rightarrow a=1$