000^0

Calculus Level 2

000^0 is often defined as equal to 11 since limx0+xx=1\displaystyle\lim_{x\to 0^+}x^x=1.

This led a mathematician to claim that limx0+f(x)g(x)=1\displaystyle\lim_{x\to 0^+}f(x)^{g(x)}=1 for all f(x)f(x) and g(x)g(x) whenever limx0+f(x)=limx0+g(x)=0\displaystyle\lim_{x\to 0^+}f(x)=\displaystyle\lim_{x\to 0^+}g(x)=0.

Is this claim true?

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