Definition of a Limit.

  1. 1. The problem statement, all variables and given/known data

    Given a function [tex]f:R\rightarrow R[/tex] and a number L,write down a definition of the statement

    [tex]\lim_{x\rightarrow-\infty}f(x)=L[/tex]


    3. The attempt at a solution

    Is it just [tex]\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow\infty}f(-x)[/tex] ?

    and definition is
    for [tex]\forall \epsilon>0[/tex] [tex]\exists N[/tex] such that [tex]\forall n>N[/tex]
    we have [tex]|f(-x)-L|<\epsilon[/tex]
     
  2. jcsd
  3. quasar987

    quasar987 4,770
    Science Advisor
    Homework Helper
    Gold Member

    assuming by n you mean x, then yes, this looks like a good dfn, although the usual dfn is that "for all e>0, there is an N<0 such that x<N ==>|f(x)-L|<e"
     
  4. Good.Thanks.
     
  5. HallsofIvy

    HallsofIvy 40,241
    Staff Emeritus
    Science Advisor

    A more "standard" definition of
    [tex]\lim_{x\rightarrow-\infty}f(x)=L[/tex]
    would be:

    "Given [itex]\epsilon> 0[/itex], there exist N such that if x< N, then [itex]|f(x)-L|<\epsilon[/itex]."

    Notice that in neither this definition nor your definition is N required to be an integer.
     
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