Math Is Fun Forum

darfmore

Member

Registered: 2012-03-02

Posts: 4

Limits - Proofs

Hey guys,

Just looking for a little guidance on this question that my friend asked me to do.
It says:

a) Suppose that lim(x⇒a)f(x) exists and is nonzero. Prove that if lim(x⇒a)g(x) does not exist, then lim(x⇒a)f(x)g(x) also does not exist.

b) Prove the same result if lim(x⇒a)f(x) = ∞.

There are two ways that I have thought to do this. One is to try and achieve a contradiction through the assumption that the limit does exist, using the epsilon/delta definition. The other is to try and prove a 'not-limit' epsilon/proof by countering the normal epsilon/delta definition. I'm not sure which of these two methods would be simpler, or if neither? Any help would be greatly appreciated.

Thanks

benice

Member

Registered: 2010-06-10

Posts: 117

Website

Re: Limits - Proofs

a)

You can prove the contrapositive:
If lim(x⇒a)f(x)g(x) exists, then lim(x⇒a)g(x) also exists.

Assume that lim(x⇒a)f(x)g(x) exists.

Since lim(x⇒a)f(x)g(x) and lim(x⇒a)f(x) both exist and lim(x⇒a)f(x) is nonzero,
it follows that lim(x⇒a)g(x) [ = lim(x⇒a)f(x)g(x)/f(x) ] exists.

Last edited by benice (2012-03-21 01:11:48)