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Mostly based on mlk's comments here
Lemma 1 $\lim_{x\to0} x^0=1$$\lim_{x\to 0^+} x^0=1$, so $0^0=1$.
Lemma 2 $\lim_{x\to0} 0^x=0$$\lim_{x\to 0^+} 0^x=0$, so $0^0=0$.
Therefore $1=0$.
Lemma 1 $\lim_{x\to0} x^0=1$, so $0^0=1$.
Lemma 2 $\lim_{x\to0} 0^x=0$, so $0^0=0$.
Lemma 1 $\lim_{x\to 0^+} x^0=1$, so $0^0=1$.
Lemma 2 $\lim_{x\to 0^+} 0^x=0$, so $0^0=0$.
