Lim $x^{(x^x)}$ as $x$ approaches zero from the right

Question

I tried this problem and I first found the lim of $x^x$ as $x$ approaches zero from right to be 1 (I did this by re-writing $x^x$ as an exponential) and when I repeated the same process to find lim of $x^{(x^x)}$, I found 1 again but the final answer should be ZERO. Could I have an explanation on why it's a zero?

Answer 1

Note that

$$x^{x^x}=e^{x^x\log x}\to0$$

indeed

$$x^x\log x\to -\infty$$

Answer 2

Using $\lim_{x\to 0^+}x^x = 1$, we have: $$\lim_{x\to 0^+}x^{(x^x)} = 0^1 = 0.$$

Answer 3

As $x\to 0^+$, we see $x^x\to 0^0$, which evidence suggests is $1$ for $x \in \Bbb R$.

Thus for $x^{x^x}$, as $x\to 0^+$ we get $0^1$, which is certainly $0$.

Incidentally, this implies that the limit of $x^{x^{x^x}}=1$, and $x^{x^{x^{x^x}}}=0$, etc...

Graphed for your perusal here