Calculate $\lim\limits_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$ Please help me solving $\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$

Since this is calculus why not try with L'Hospital?
$$\lim_{x\to a}\frac{a^{a^x}-a^{x^a}}{a^x-x^a}=\lim_{x\to a}\frac{a^xa^{a^x}\log^2a-ax^{a-1}\log a\cdot a^{x^a}}{a^x\log a-ax^{a-1}}=$$
$$=\frac{a^aa^{a^a}\log^2a-a^a\log a\cdot a^{a^a}}{a^a\log a-a^a}=\frac{a^{a^a}\log^2a-\log a\cdot a^{a^a}}{\log a -1}=a^{a^a}\log a$$
