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 June 23rd, 2011, 11:56 AM #1 Newbie   Joined: Jun 2011 Posts: 4 Thanks: 0 Prove a property in limits How can I prove that the equality: $lim(f(x)^r)=[limf(x)]^r$
June 23rd, 2011, 12:42 PM   #2
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Re: Prove a property in limits

Quote:
 Originally Posted by harkaz2 How can I prove that the equality: $lim(f(x)^r)=[limf(x)]^r$
Use this rule:
Assume that both $f(x)$ and $g(x)$ are defined near $a$ and $\lim_{x \rightarrow a}{f(x)}=F,\quad \lim_{x \rightarrow a}{g(x)}=G$ then $\lim_{x \rightarrow a}{f(x) \cdot g(x)}=F\cdot G$
Then $\lim_{x \rightarrow a}{f(x)^r}=F^r$, if $r$ is a positive integer.

 June 24th, 2011, 02:20 AM #3 Newbie   Joined: Jun 2011 Posts: 4 Thanks: 0 Re: Prove a property in limits Well, I have proven that the statement is true for every rational r. How can I generalise for every r in R?
June 24th, 2011, 06:32 AM   #4
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Re: Prove a property in limits

Quote:
 Originally Posted by harkaz2 Well, I have proven that the statement is true for every rational r. How can I generalise for every r in R?
Won't you get problems for negative r? Example:

$\lim_{n\to \infty } \left(\frac{1}{n}\right)^{-2}=\infty$
$[\lim_{n\to \infty } \left(\frac{1}{n}\right)]^{-2}$ is undefined.

 June 24th, 2011, 11:19 AM #5 Newbie   Joined: Jun 2011 Posts: 4 Thanks: 0 Re: Prove a property in limits The problem you mention is not my case. I am talking about this category of limits (I didn't know how to put the x-> c notation under the lim symbol in the original post) : $\lim_{x \rightarrow c}{g(x)}$ where c is real number. The case you mention is when x goes to infinity, somewhat different There is no doubt that the property is true; I have seen it in various textbooks. The problem is how I prove it. Should I use dedekind cuts? Or density of rational numbers is enough to prove it? If anyone could help completing this part of the proof, I would much appreciate it.
 June 26th, 2011, 11:34 AM #6 Newbie   Joined: Jun 2011 Posts: 4 Thanks: 0 Re: Prove a property in limits Proof found! Topic is thus closed.

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