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Math Help - Existence of Limit

  1. #1
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    Existence of Limit

    Let 0 \leq f(x)
    Then proof existence of infinity or a limit (but not both):
    Thus,
    \lim c \rightarrow \infty f(x)=L
    or
    \lim c \rightarrow \infty f(x)=\infty

    Logic caution:
    In the problem you have to proof that exactly one of the conditions must be satisfied and exactly one.
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  2. #2
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    Quote Originally Posted by ThePerfectHacker
    Let 0 \leq f(x)
    Then proof existence of infinity or a limit (but not both):
    Thus,
    \lim c \rightarrow \infty f(x)=L
    or
    \lim c \rightarrow \infty f(x)=\infty

    Logic caution:
    In the problem you have to proof that exactly one of the conditions must be satisfied and exactly one.
    Let f(x)=\sin(x)+1, then what is (if anything):

    \lim_{x \rightarrow \infty}f(x)\ ?

    Or have I misunderstood your intention?
    Last edited by CaptainBlack; December 25th 2005 at 01:26 PM.
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  3. #3
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    Yes, I made a mistake with what I said you are correct.
    Show that if the limit is L then it cannot be infinite.
    Show that if the limit is infinite then it cannot be L.
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  4. #4
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    Quote Originally Posted by ThePerfectHacker
    Yes, I made a mistake with what I said you are correct.
    Show that if the limit is L then it cannot be infinite.
    Show that if the limit is infinite then it cannot be L.

    in case of infinity then for any given M there is X that for every x>X, f(x)>M
    take M to be L+100 and L is not a limit

    in case of L for every given g>0 there is X wich for every x>X, |f(x)-L|<g
    take X to be the one that from him forward f(x) is blocked (don't remember the english expression) (and don't remember the proof that you have one) in this part f(x)<>infinity
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