Why does this limit exist $x^{x}$

I was wondering what is the value of $\lim\limits_{x\to 0+} x^{x}$.

Assuming that the limit exists, I could show using the usual logarithm techniques that the limit is $1$. However, I am not able to show that the limit exists. Could some one help on that?

EDIT: It would be so nice if we could do it without l'Hopital...

Solution 1:

Hint: Using the change of variable $x=e^{-t}$ show that $$\lim_{x \to 0+} x \ln x = 0.$$

$$\lim_{x \to 0+} x \ln x = \lim_{t \to +\infty} -\frac{t}{e^t} = 0.$$

To prove the last equality you can use the Taylor expansion for $e^t$, for instance.

Solution 2:

By the l'Hôpital theorem we have

$$\lim_{x\to0}x\log x=\lim_{x\to0}\frac{\log x}{\frac 1 x}=-\lim_{x\to0}\frac{\frac 1 x}{\frac{ 1 }{x^2}}=-\lim_{x\to0}x=0$$

and conclude with $$x^x=e^{x\log x}$$

Edited To find the limit without the l'Hôpital theorem: for $0<x<1$: $$ 0\leq|x\log x|=x\int_x^1\frac{dt}{t}\leq x\int_x^1\frac{dt}{t^{3/2}}=-2x\frac{1}{\sqrt{t}}\Big|_x^1\to0$$

Solution 3:

Assume you know $$\lim_{n \rightarrow \infty}\sqrt[n]{n}=1$$ Proof:

$$1 \leq \sqrt[n]{n}<\frac{n-2+2\sqrt n}{n}, \,\,\,\,\,\,\,\,\,\, \mathrm{GM \leq AM}$$

Now for each $x$ near zero define $n(x)=[\frac{1}{x}]$ where $[\cdot]$ is the integral part function. Let's call $n(x)$ out of simplicity $n$.

Therefore \begin{align} &\frac{1}{n} \geq x \geq \frac{1}{n+1} \frac{}{} \Rightarrow\\ \left (\frac{1}{n} \right )^\frac{1}{2n}\geq &\left (\frac{1}{n} \right )^{\frac{1}{n+1}} \geq x ^{x}\geq \left (\frac{1}{n+1} \right )^\frac{1}{n} \geq \left (\frac{1}{2n} \right )^\frac{1}{n}& \end{align}

And because as $x$ tends to zero $n(x)$ tends to infinity taking limits at both sides you are done.