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.