\(0^0\)

Calculus Level 1

\(0^0\) 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?

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