limx→a xa−axxx−aa is equal to

# $\underset{x\to a}{lim} \frac{{x}^{a}-{a}^{x}}{{x}^{x}-{a}^{a}}$ is equal to

1. A

$\frac{1+{\mathrm{log}}_{e}a}{1-{\mathrm{log}}_{e}a}$

2. B

$\frac{{\mathrm{log}}_{e}\left(e/a\right)}{{\mathrm{log}}_{e}\left(ae\right)}$

3. C

$\frac{{\mathrm{log}}_{c}\left(a/e\right)}{{\mathrm{log}}_{e}\left(ae\right)}$

4. D

none of these

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### Solution:

We have,

$\underset{x\to a}{lim} \frac{{x}^{a}-{a}^{x}}{{x}^{x}-{a}^{a}}$

$=\underset{x\to a}{lim} \frac{a{x}^{a-1}-{a}^{x}\mathrm{log}a}{{x}^{x}\left(1+\mathrm{log}x\right)-0}$                       [Using L' Hospital's Rule]

$=\frac{{a}^{a}-{a}^{a}\mathrm{log}a}{{a}^{a}\left(1+\mathrm{log}a\right)}=\frac{1-\mathrm{log}a}{1+\mathrm{log}a}=\frac{{\mathrm{log}}_{e}\left(e/a\right)}{{\mathrm{log}}_{e}\left(ae\right)}$

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