Definition of a Limit.


by azatkgz
Tags: definition, limit
azatkgz
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Oct21-07, 08:50 AM
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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]
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quasar987
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Oct21-07, 10:24 AM
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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"
azatkgz
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Oct21-07, 10:25 AM
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Good.Thanks.

HallsofIvy
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Oct21-07, 12:40 PM
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Definition of a Limit.


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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