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Precise Definition of a Limit at Negative Infinity

  1. Aug 24, 2014 #1
    I'm working through some problems from Stewart's Calulus, 6ed. and am having some difficulty with certain limit proofs. In particular, there is no definition provided for limits of the form:

    $$ \lim_{x \to - \infty} f(x) = L $$

    One of the exercises is to come up with a formal definition for such a proof - which I did and used successfully a few times about a month ago when I was working through that section - but I want to make sure that my definition is really correct and not just lucky for those two problems.


    I would suspect the definition of such a limit to be something like:

    We say

    $$ \lim_{x \to - \infty} f(x) = L $$

    if for every ##\epsilon>0## there exists ##N## such that ##x < N \implies |f(x) - L| < \epsilon##.


    Is this the correct precise definition of such a limit? I was unable to find an answer browsing google - most websites seem to provide the precise definition for "normal" limits and for limits at positive infinity but not at negative infinity.
     
  2. jcsd
  3. Aug 24, 2014 #2

    micromass

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    Sounds good.
     
  4. Aug 24, 2014 #3
    Thank you for your prompt reply!
     
  5. Aug 24, 2014 #4

    BvU

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    Would sound even better if you had written ##x > N##, but I'm pretty sure the < was just a typo...
     
  6. Aug 24, 2014 #5

    vela

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    The OP wrote it correctly because the limit is for ##x## going to ##-\infty##.
     
  7. Aug 24, 2014 #6

    BvU

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    My apologies. Couldn't imagine such a quirk; proves how rusty one can get with old age... :redface:
     
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