Example of limit of indeterminate form $0^0$ where the limit is not equal to $1$?

Question

I know of one example: $\lim_{x\rightarrow 0^+} 0^x=0$, not $1$. But are there any other, more interesting examples? Every example I cook up seems to have a limit of $1$.

Answer 1

Yes indeed since for $x\neq 0$ we have $0^x=0$

$$\lim_{x\rightarrow 0} 0^x=\lim_{x\rightarrow 0} 0=0$$

more in general to have

$$\lim_{x\rightarrow 0^+} f(x)^x=0$$

since

$$f(x)^x=e^{x\log(f(x))}$$

it suffices that $x\log(f(x))\to -\infty$ to construct such kind of examples.

Answer 2

For instance:

• $\lim_{x\to 0} \left(e^{-x^{-2}}\right)^{\lvert x\rvert}=0$
• $\lim_{x\to 0}\left(\exp\frac{1}{-2\lvert x\rvert}\right)^{\lvert x\rvert}=e^{-1/2}$