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A statement equivalent to the definition of limits at infinity?

  1. Mar 10, 2013 #1
    I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition.

    So the question is this: are the following two statements equivalent?

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

    (2) [tex]\exists c>0\exists M>0\left(\sup\left\{ \left|x\left(f\left(x\right)-L\right)\right|:x\geq c\right\} \leq M\right)[/tex]
  2. jcsd
  3. Mar 10, 2013 #2
    Consider [itex]f(x)=\frac{1}{\log(x)}[/itex] and [itex]L=0[/itex].
  4. Mar 10, 2013 #3


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    Just to nitpick a bit: you're thinking of the limit _as x tends to infinity_, and not quite the limit _at infinity_ , since infinity is not a real number ( at least not a standard real ).
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