- #1
Hobold
- 83
- 1
Homework Statement
Make [itex]f, g : X \subseteq \mathbb{R} \rightarrow \mathbb{R}[/itex] function with [itex]g[/itex] being a limited function and [itex]\lim_{x \to a} f(x) = 0[/itex] for [itex]a \in X[/itex]. Prove that [itex]\lim_{x \to a} f(x)g(x) = 0[/itex].
Homework Equations
A function [itex]g[/itex] is limited if there's a [itex]M>0[/itex] for which [itex]|g(x)| >= M[/itex]
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.