Proof of power rule of limit laws

[burkley]

Homework Statement

Power Rule: If r and s are integers with no common factor and s=/=0, then
lim(f(x))^r/s = L^r/s
x$$\rightarrow$$c
provided that L^r/s is a real number. (If s is even, we assume that L>0)
How can I prove it?

Homework Equations

The Attempt at a Solution

I heard that the proof is related to The Sandwich Theorem

 

[HallsofIvy]

From $\lim_{x\rightarrow c} f(x)g(x)= \left(\lim_{x \rightarrow c}f(x)\right)\left(\lim_{x \rightarrow c}g(x)$ you should be able to prove that $\lim_{x\rightarrow c}f^2(x)= \left(\lim_{x\rightarrow c} f(x)\right)^2$. Then use induction to prove that $\lim_{x\rightarrow c}f^n(x)= \lim_{x\rightarrow c}\left(f(x)\right)^n$. That's the easy part.

For $\lim_{x\rightarrow c} f^{1/m}(x)$, assuming that limit exists, define $g(x)= f^{1/n}(x)$ and look at $lim_{x\rightarrow c}g^n(x)$.

 

[burkley]

Thank you for answering me. But how can we assume that lim f^1/m(x) exists?
x->c