First slide

Introduction to limits

Question

$lim_{x \to a} \frac{x^{a} - a^{x}}{x^{x} - a^{a}}$ is equal to

Moderate

A

$\frac{1 + \log_{e} a}{1 - \log_{e} a}$
B

$\frac{\log_{e} (e / a)}{\log_{e} (a e)}$
C

$\frac{\log_{c} (a / e)}{\log_{e} (a e)}$
D

none of these

Solution

We have,
$lim_{x \to a} \frac{x^{a} - a^{x}}{x^{x} - a^{a}}$ $[\frac{0}{0} f o r m]$
$= lim_{x \to a} \frac{a x^{a - 1} - a^{x} \log a}{x^{x} (1 + \log x) - 0}$ [Using L' Hospital's Rule]
$= \frac{a^{a} - a^{a} \log a}{a^{a} (1 + \log a)} = \frac{1 - \log a}{1 + \log a} = \frac{\log_{e} (e / a)}{\log_{e} (a e)}$

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