# Thread: help proving lim question

1. ## help proving lim question

hello you all, i'm new here.. i'd like if you help me in this question.

f and g are function defined in a punctured neighborhood of X0, and L is a real number.
assuming lim(x->x0) (f times g)(x)=L

i need to prove\disprove that if lim(x->x0) f(x)=infinity, then lim(x->x0) g(x) exists.

it would be very helpful if you also instruct me in the general way of proving such questions, because i'm kinda new in this subject...
thx!

2. ## Re: help proving lim question

Are you allowed to take for granted that the limit of a product is equal to the product of limits?

3. ## Re: help proving lim question

Originally Posted by Prove It
Are you allowed to take for granted that the limit of a product is equal to the product of limits?
sorry, but i didn't understand you..

4. ## Re: help proving lim question

Prove It was asking if you have the theorem that says "if $lim_{x\to a} f(x)= F$ and $\lim_{x\to a}g(x)= G$ then $\lim_{x\to a} f(x)g(x)= FG$. However, that does not appear to me to apply here since $\lim_{x\to x_0} f(x)$ does not exist. Instead, it should be apparent that, in order that $\lim_{x\to x_0} f(x)g(x)$ exist, the limit of g(x) must not only exist but must be a very specific number!

5. ## Re: help proving lim question

Originally Posted by HallsofIvy
Prove It was asking if you have the theorem that says "if $lim_{x\to a} f(x)= F$ and $\lim_{x\to a}g(x)= G$ then $\lim_{x\to a} f(x)g(x)= FG$. However, that does not appear to me to apply here since $\lim_{x\to x_0} f(x)$ does not exist. Instead, it should be apparent that, in order that $\lim_{x\to x_0) f(x)g(x)$ exist, the limit of g(x) must not only exist but must be a very specific number!
why does it clear that in order the limit of f(x)*g(x) to be L, there's must to be a limit of g(x)? i didn't get that... how do you prove that?

6. ## Re: help proving lim question

The hypothesis is that $\lim_{x\to x_0} f(x)$ is "infinity" which means that f(x) is very very large close to $x_0$. So what happens to f(x)g(x)? What must be true of g(x) so that f(x)g(x) does NOT get "very very large" close to $x_0$.

7. ## Re: help proving lim question

but can we multiple infinity by 0? i mean, if i got you right, you implied that in order that f(x)g(x) doesn't get very large (and to be L, a real number, as it's given), we need something to balance it -> lim g(x). and as i see it, it can only be done if lim g(x) exsits and equal 0, so then lim f(x)g(x) will equal 0 too.
but if i'm not wrong, you can't multiple infinty by 0, can we?