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Limited function

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data

    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].


    2. Relevant equations

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

    3. 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.
     
  2. jcsd
  3. May 3, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    do you mean that [itex]|g(x)| \leq M[/itex]?

    knowing M couldn't you choose x close enough to a, such that f(x) is much smaller than M?
     
    Last edited: May 3, 2010
  4. May 3, 2010 #3

    Mark44

    Staff: Mentor

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