I'm looking for a proof of the following statement:

If lim f(x) exists, then lim nth root [ f(x) ] = nth root [ lim f(x) ]

I can't seem to find a proof in any Calculus/Analysis books.

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- Oct 22nd 2011, 06:58 AMAlfieLimit of the nth root of a function
I'm looking for a proof of the following statement:

If lim f(x) exists, then lim nth root [ f(x) ] = nth root [ lim f(x) ]

I can't seem to find a proof in any Calculus/Analysis books. - Oct 22nd 2011, 07:16 AMemakarovRe: Limit of the nth root of a function
If $\displaystyle \lim f(x) > 0$, then this holds by the chain rule (the statement about the limit of composition) since $\displaystyle \sqrt[n]{x}$ is continuous for $\displaystyle x>0$.