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Math Help - Limits and continuity

  1. #1
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    Limits and continuity

    Limits and continuity-question-3-assignement-3.pngany help with this question would be muchly appreciated
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  2. #2
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    Re: Limits and continuity

    How rigorous does this need to be? For instance, are you suppose to use the epsilon-delta definitions? It is very easy to apply in this case:

    For all \epsilon >0, choose \delta = \frac{1}{2} so that 0<|x-3|<\frac{1}{2} \implies x\not \in \mathbb{Z} \implies f(x)=2 \implies |f(x)-2|=0 < \epsilon. So the limit is  2.

    You can easily generalise this to arbitrary a\in \mathbb{Z} instead of 3. And by generalise I mean replace every instance of 3 with a and you'll get a valid proof.

    The limit for b) is also two. You need minor modification to the proof above. For example, take \delta to be the distance for \pi (or a\not \in \mathbb{Z}) to the nearest integer.

    In light of parts a) and b), it should be obvious if and where the function is continuous.
    Last edited by Gusbob; April 17th 2013 at 02:35 AM.
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  3. #3
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    Re: Limits and continuity

    For x any number "close" to 3, but not equal to 3, x is NOT an integer so f(x)= 2. Therefore \lim_{x\to 3}f(x)= ?

    For x any number "close" to \pi, but not equal to \pi, x is NOT an integer so f(x)= 2. Therefore \lim_{x\to 3}f(x)= ?
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