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Math Help - Absolute value of the limit of a function is equal to the absolute value of the limit

  1. #1
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    Absolute value of the limit of a function is equal to the absolute value of the limit

    The question:

    Let f: D-> R and let c be an accumulation point of D. Suppose that lim_x->c f(x)=L.

    Prove that lim_x->c |f(x)|=|L|.

    A sketch of what I have:
    Definition of limit and then

    |f(x)-L|< \varepsilon

    ||f(x)|-|L|| \leq |f(x)-L|< \varepsilon

    But I'm sketchy on the reasons and if this is the right direction. Any ideas? Thanks.
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  2. #2
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    Re: Absolute value of the limit of a function is equal to the absolute value of the l

    Quote Originally Posted by CountingPenguins View Post
    The question:

    Let f: D-> R and let c be an accumulation point of D. Suppose that lim_x->c f(x)=L.

    Prove that lim_x->c |f(x)|=|L|.

    A sketch of what I have:
    Definition of limit and then

    |f(x)-L|< \varepsilon

    ||f(x)|-|L|| \leq |f(x)-L|< \varepsilon

    But I'm sketchy on the reasons and if this is the right direction. Any ideas? Thanks.
    Just throw in those \deltas from the definition of limit and you have your proof
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