$0^0$ is often defined as equal to $1$ since $\displaystyle\lim_{x\to 0^+}x^x=1$ .

This led a mathematician to claim that $\displaystyle\lim_{x\to 0^+}f(x)^{g(x)}=1$ for all $f(x)$ and $g(x)$ whenever $\displaystyle\lim_{x\to 0^+}f(x)=\displaystyle\lim_{x\to 0^+}g(x)=0$ .

Is this claim true?

No
Yes

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The mathematician August Ferdinand Möbius made the claim.

However, it is said that a commentator who anonymously signed himself as "S" gave a counterexample when $f(x)=e^{-1/x}$ and $g(x)=x$ .