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.