Proving \lim_{x \to a} f(x)g(x) = 0

[Hobold]

Homework Statement

Make f, g : X \subseteq \mathbb{R} \rightarrow \mathbb{R} function with g being a limited function and \lim_{x \to a} f(x) = 0 for a \in X. Prove that \lim_{x \to a} f(x)g(x) = 0.

Homework Equations

A function g is limited if there's a M>0 for which |g(x)| >= M

The Attempt at a Solution

It seems pretty obvious that the affirmation is true, but I can1t find a proof for that. You can't also assume that g has a limit at x -> a because there's nothing saying that, therefore it's not possible to use a direct proof by limits properties.

I've been trying to do this by definition of limits, but I always get that limit of g when x tends to a has to exist, which is not true.

 

[lanedance]

do you mean that |g(x)| \leq M?

knowing M couldn't you choose x close enough to a, such that f(x) is much smaller than M?

 

[Mark44]

A function g is limited if there's a M>0 for which |g(x)| >= M

The usual terminology for such a function, with lanedance's correction, is bounded, not limited.