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# With great power, comes great responsibility.

Can you prove that $$0^0=1$$ ?

Note by Bruce Wayne
3 years, 11 months ago

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Hi this question has been asked repeatedly on brilliant,

$$0^0 = e^{0 \ln 0}$$ is undefined , but we can find $$\displaystyle \lim_{x \to 0} x^x = \lim_{x \to 0} e^{x \ln x } = 1$$ , as $$\displaystyle \lim_{x \to 0} x\ln x = 0$$ ,

This is very similar to saying that $$\frac{0}{0}$$ is undefined but $$\displaystyle\lim_{x \to 0 } \frac{x}{x}$$ is defined.

Also $$\displaystyle\lim_{(f(x) \to 0, g(x) \to 0} {\big(f(x)\big)}^{g(x)}$$ is not $$1$$ , it would depend on $$f(x)$$ and $$g(x)$$

- 3 years, 11 months ago

- 3 years, 11 months ago

0^0 is undefined...

- 3 years, 11 months ago