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Definition of a limit

  1. Apr 20, 2015 #1
    1. The problem statement, all variables and given/known data
    It is not exactly a homework question, but why does the definition of a limit use strict inequalities as follows:
    if 0 < |x - a| < δ, then |f(x) - l| < ε
    rather than weak inequalities, for example
    if 0 < |x - a| < δ, then |f(x) - l| ≤ ε

    Could the addition of the equality option make a difference?

    2. Relevant equations


    3. The attempt at a solution
    I tried thinking of functions that would yield different limits to the limit produced by the formal definition, but couldn't find any.
    I also tried to rule it out somehow with formal deduction, but couldn't.
    Any hints or ideas?

    Thanks
     
  2. jcsd
  3. Apr 20, 2015 #2
    Ok, I realized that the fact I couldn't disprove it is because it indeed holds.
    It might not be as nice to be the definition, but given the definition,
    if 0 < |x - a| < δ, then |f(x) - l| ≤ ε
    implies that the limit of f is l.
     
  4. Apr 20, 2015 #3
    We proved early in our analysis course as a "recreational" activity that the two are, in fact, equivalent statements, but we just agreed to use the strict inequality.
     
  5. Apr 20, 2015 #4

    Fredrik

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    It's obvious that if ##\lim_{x\to a}f(x)=l## in the sense of the standard definition, then ##\lim_{x\to a}f(x)=l## in the sense of the alternative definition.

    Suppose that ##\lim_{x\to a}f(x)=l## in the sense of the alternative definition. Let ##\varepsilon>0##. Let ##\delta>0## be such that the following implication holds for all ##x\in\mathbb R##,
    $$0<|x-a|<\delta~\Rightarrow~|f(x)-l| <\frac\varepsilon 2.$$ (Our assumption ensures that such a ##\delta## exists). For all ##x\in\mathbb R## such that ##0<|x-a|<\delta##, we have ##|f(x)-l|<\frac\varepsilon 2<\varepsilon##. This implies that ##\lim_{x\to a}f(x)=l## in the sense of the standard definition.
     
  6. Apr 23, 2015 #5
    Thanks for your responses.
    Could you refer me to the analysis course you mentioned?

    Edit: Oh, I guess you meant a course you took elsewhere, not some section here in PF. Nvm...
     
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