limit identity proofs

**Greengoblin** · December 9th 2008, 11:33 AM

I have these identities for limits:

let, $\lim_{x\to a} f(x)=b\text{ }\lim_{x\to a} g(x)=c$

then:

$\textcircled1 \lim_{x\to a}kf(x)=kb\text{ for a fixed constant k}$

$\textcircled2 \lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x)=b\pm c$

$\textcircled3 \lim_{x\to a}[f(x)g(x)]=\lim_{x\to a}f(x)\lim_{x\to a}g(x)=bc$

$\textcircled4 \lim_{x\to a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}=\frac{b}{c} \text{ } (c\ne 0)$

$\textcircled5 \lim_{x\to a}[f(x)]^{\frac{1}{n}}=b^{\frac{1}{n}} \text{ if } [f(x)]^{\frac{1}{n}} \text{ and } b^{\frac{1}{n}} \text{ are real}$

Apparently there is some rigorous way of proving these, but my book doesn't give them. Can someone give me some hints of what I need to do? Do I use the Epsilon-delta method msomehow?

(please don't prove, I want to have a crack myself).

**galactus** · December 9th 2008, 12:06 PM

Here is a nice one on the product of limits. Yes, you can use the epsilon-delta method to show them. If you want rigor, that would be the way to go.

http://www.math.uchicago.edu/~ershov...oduct_rule.pdf

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