Evaluating $0^0$ and its limit

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Zero to zero power

From what I understand $0^0$ is indeterminate, yet when you evaluate $\lim\limits_{x\to 0}x^0$ you get 1 (given on wolframalpha.com). Something seems wrong about this.

If wolframalpha is correct, how can this be?

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Your limit is correct. The trouble is that is not the only way to approach $0^0$.

Consider $0^x$, $x>0$. Then the limit as $x$ approaches $0$ from above is equal to $0$. For the limit to exist, every pair of functions $f(x), g(x)$, with both functions approaching $0$ as $x \to 0$, must agree on $$\lim_{x \to 0} f(x)^{g(x)}$$

and these two examples show there are distinct pairs of functions which disagree on the limit.

(in this context, we might restrict to $f, g$ being nonnegative functions, but the same principle applies)