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Math Help - real analysis one sided limits

  1. #1
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    real analysis one sided limits

    hey guys. so im in an introductory real analysis course and I am finding it quite challenging, partially because my teacher isnt great. i went to ask for help on some limit problems but im not sure if his advice was accurate.

    1) Find lim(x approaches x0-) f(x) and lim((x approaches x0+) f(x) for f(x)=(x+abs(x))/x x0=0 i found the solutions (2, x>0 and 0, x<0) easily. The book asks for an epsilon-delta proof if possible and my teacher claims it is trivial because they are constant. if this is the case when should you finish the proof?


    2) I would also appreciate help with a second problem that asks the same thing except when f(x)= abs(x-1)/(x2 +x-2), x0=1

    3) prove: if limf(x) exists, there is a constant M and a p>0 such that abs(f(x)) < or = M if 0<abs(x-x0)<p. I was having trouble with this one also. in this case, basically p is delta right? and if we assume f(x)=M, then abs(M-L)=epsilon. should i approach this proof with a contradiction?

    thanks for all advice!
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  2. #2
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    Re: real analysis one sided limits

    Quote Originally Posted by biga415 View Post
    1) Find lim(x approaches x0-) f(x) and lim((x approaches x0+) f(x) for f(x)=(x+abs(x))/x x0=0 i found the solutions (2, x>0 and 0, x<0) easily. The book asks for an epsilon-delta proof if possible and my teacher claims it is trivial because they are constant. if this is the case when should you finish the proof?

    2) I would also appreciate help with a second problem that asks the same thing except when f(x)= abs(x-1)/(x2 +x-2), x0=1

    3) prove: if limf(x) exists, there is a constant M and a p>0 such that abs(f(x)) < or = M if 0<abs(x-x0)<p. I was having trouble with this one also. in this case, basically p is delta right? and if we assume f(x)=M, then abs(M-L)=epsilon. should i approach this proof with a contradiction?
    I for one find this almost impossible to read!
    It looks like the first question is:
    \lim _{x \to 0^ -  } \frac{{x + |x|}}{x}\;\& \,\lim _{x \to 0^ +  } \frac{{x + |x|}}{x}

    If you click on the "reply with quote" tab you will have access to the LaTeX code.
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  3. #3
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    Re: real analysis one sided limits

    haha, sorry, i dont know how to use LaTex, however, you did get the problem correct!
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    Re: real analysis one sided limits

    #2
    f(x)=\frac{{|x-1|}}{}{x^{2}+x-2}
    Last edited by biga415; October 2nd 2012 at 07:44 PM.
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  5. #5
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    Re: real analysis one sided limits

    Quote Originally Posted by biga415 View Post
    dont know how to use LaTex, however, you did get the problem correct!
    \frac{{x + \left| x \right|}}{x} = \left\{ {\begin{array}{rr}   {0,} & {x < 0}  \\   {2,} & {x > 0}  \\\end{array}} \right.
    This makes #1 easy.
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  6. #6
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    Re: real analysis one sided limits

    For #2, factor the denominator, then try to do what Plato did in the 1st example. When x>1, what's f(x)? When x<1, what's f(x)? Once you understand that, determining the two one-sided limits will hopefully be easy.
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  7. #7
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    Re: real analysis one sided limits

    One of my main questions was whether or not I need to do the epsilon-delta proof at all for number one. my professor claims i dont, but if thats the case when should I? also would i show two one-sided limit proofs?
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  8. #8
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    Re: real analysis one sided limits

    Yes, I think that's what he's asking for, an epsilon-delta proof for each of the one-sided limits.

    If you recall, the full limit speaks of \forall x s.t. 0 < | x - a | < \delta, which is \forall x \in (a-\delta, a) \cup (a, a+\delta).

    The one sided limits are defined exactly the same, except applying to only one of those halves:

    For \lim_{x \to a^-} it's \forall x \ s.t. \ a - \delta < x < a, which is \forall x \in (a-\delta, a).

    For \lim_{x \to a^+} it's \forall x \ s.t. \ a < x< a + \delta, which is \forall x \in (a, a+\delta).

    In full:

    \lim_{x \to a^-}f(x) = L iff \forall \epsilon > 0 \ \exists \delta > 0 \ni a - \delta < x < a \Rightarrow \lvert f(x) - L \rvert < \epsilon.

    \lim_{x \to a^+}f(x) = L iff \forall \epsilon > 0 \ \exists \delta > 0 \ni a < x < a+ \delta \Rightarrow \lvert f(x) - L \rvert < \epsilon.
    Last edited by johnsomeone; October 3rd 2012 at 06:01 AM.
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