If limx→a ax−xaxx−aa=−1, then the value of a is

A
1
B
0
C
$e$
D
none of these

Solution:

We have,

$lim_{x \to a} \frac{a^{x} - x^{a}}{x^{x} - a^{n}} = - 1$

$\Rightarrow lim_{x \to a} \frac{a^{x} \log a - a x^{a - 1}}{x^{x} (1 + \log x)} = - 1$ [Using L' Hospital's Rule]

$\begin{array}{l} \Rightarrow \frac{a^{a} \log_{e} a - a \cdot a^{a - 1}}{a^{a} (1 + \log_{e} a)} = - 1 \\ \Rightarrow \frac{\log_{e} a - 1}{\log_{e} a + 1} = - 1 \Rightarrow \log_{e} a = 0 \Rightarrow a = 1 \end{array}$

If $lim_{x \to a} \frac{a^{x} - x^{a}}{x^{x} - a^{a}} = - 1,$ then the value of a is

Solution:

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