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Math Help - Epsilon delta definition of limit.

  1. #1
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    Epsilon delta definition of limit.

    Q1) The precise definition of lim x->a (f(x)≠L) is as follows

    There exists an ε>0 such that for every δ>0 and if 0<|x-a|<δ then |f(x)-L|>ε

    Show that lim x->0 sin(pi/x)≠L

    note: Take L=0.19


    Q2) Calculate lim x->1 (sqrt(x+8)) and prove it by using ε and δ definition of limit.
    Last edited by mr fantastic; October 20th 2011 at 01:39 PM. Reason: Title.
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  2. #2
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    Re: Plz help me about epsilon delta definition of limit.

    Welcome to MHF.

    Show us what you have tried.

    Where are you stuck?
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  3. #3
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    Re: Plz help me about epsilon delta definition of limit.

    Thank you!
    about Q2 I find 0<|x-1|<δ and |sqrt(x+8)-3|<ε and |(sqrt(x+8)-3)*(sqrt(x+8)+3)/(sqrt(x+8)+3)|<ε equals |(x-1)/(sqrt(x+8)+3)|<ε Am I right


    But about Q1 I can not find anything.
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  4. #4
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    Re: Plz help me about epsilon delta definition of limit.

    Q2: That's a good beginning, to write it more formal, we have to show that:
    \forall \epsilon>0, \exist \delta>0, \forall x \in \ \mbox{dom f}: 0<|x-1|<\delta \Rightarrow |\sqrt{x+8}-3|<\epsilon
    Proof:
     |\sqrt{x+8}-3|<\epsilon \Rightarrow \left|\frac{(\sqrt{x+8}-3)(\sqrt{x+8}+3)}{\sqrt{x+8}+3}\right|\Rightarrow \left|\frac{x-1}{\sqrt{x+8}+3}\right| \Rightarrow \frac{|x-1|}{|\sqrt{x+8}+3|}<\epsilon
    Now, suppose \delta=1 therefore |x-1|<1 \Leftrightarrow -1<x-1<1 \Leftrightarrow 0<x<2 ... go further with this given to find for each \epsilon a corresponding \delta
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  5. #5
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    Re: Epsilon delta definition of limit.

    Thank you. What about Q3?
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  6. #6
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    Re: Epsilon delta definition of limit.

    Quote Originally Posted by mervecetin View Post
    Q1) The precise definition of lim x->a (f(x)≠L) is as follows

    There exists an ε>0 such that for every δ>0 and if 0<|x-a|<δ then |f(x)-L|>ε

    Show that lim x->0 sin(pi/x)≠L

    note: Take L=0.19
    For Q1:

    Start by inserting the given quantities, in this case L, a, and f(x), into the precise definition for the limit not existing.

    Please, show us something.
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