Question

If ${lim}_{x\to a}\frac{a^x-x^a}{x^x-a^a}=-1$ find $a$.

Answer 1

$\underset{x\to a}{lim}\frac{a^x-x^a}{x^x-a^a}=-1$

By $L^1-H$

$\Rightarrow \underset{x\to a}{lim}\frac{a^{x}lna-a{x}^{a-1}}{x^x(1+lnx)}=-1$

$\Rightarrow \frac{a^{a}lna-a\cdot a^{a-1}}{a^a(1+lna)}=-1$

$\Rightarrow \frac{a^a(lna-1)}{a^a(1+lna)}=-1$

$\therefore lna-1=-1-lna$

$=2lna=0$

$\therefore a=1$