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]
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"
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.