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Thread: 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|<$\displaystyle \varepsilon$

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

    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|<$\displaystyle \varepsilon$

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

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