Find the limit $\lim_{x \to 0^{+}} (x^{x^{x}} - x^x)$

Question

How do we find $$\lim_{x \to 0^{+}} (x^{x^{x}} - x^x)??$$ The answer given is equal to $-1$. Any help is appreciated.

Answer 1

Write $$ x^{x^x}=e^{x^x\log x}=e^{e^{x\log x}\log x}$$ and

$$x^x=e^{x\log x}$$ thus the limit become

$$\lim_{x\rightarrow 0^+}\Big( e^{e^{x\log x}\log x}\space -\space e^{x\log x}\Big)=\lim_{x\rightarrow 0^+}e^{e^{x\log x}\log x}-\lim_{x\rightarrow 0^+}e^{x\log x}$$

Now it easy to see that the first limit is $0$ and second is $1$; thus the different is exactly $-1$.