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I'm working through Spivak's Calculus book which proved the following:

$$\lim_{x \to a}\ (f+g)(x) = \lim_{x \to a}\ f(x) + \lim_{x \to a}\ g(x)$$ $$\lim_{x \to a} \ (f \cdot g)(x) = \lim_{x \to a}\ f(x) \cdot \lim_{x \to a}\ g(x)$$ $$\lim_{x \to a} \ \Bigg( \frac{1}{g} \Bigg) (x) = \frac{1}{\lim \limits_{x \to a} \ g(x)}$$

However, the proof that $$\lim_{x \to a}[f(x)^\alpha] = \left[\lim_{x \to a}f(x) \right]^\alpha$$ where $\alpha$ is a real number is missing.

It's easy to prove from the above properties when $\alpha$ is an integer, but what about otherwise? I've looked online and only found proofs when it is an integer.

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    More general power rules require you to take care about the sign of $f(x)$ and how you define $y^\alpha$. For arbitrary $\alpha$ and $y>0$, this tends to be defined as $y^\alpha = e^{\alpha \log y}$. But that is stepping outside of beginning calculus, so it tends to not be covered when you are first learning limits.2011-11-18

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