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

means that for every [imath]\epsilon > 0[/imath], there corresponds a positive number N, such that

[imath]\left | f(x)-L \right | < \epsilon[/imath] whenever x>N

{bolding added}

Now, one question that I have is that is there any reason to specify that N must be a positive number. Consider the function

[math]f(x)=\frac{1}{x+3}[/math]

Now, if epsilon=.2 then N=2; epsilon=.25, N=1 -- but then; epsilon=.5, N=-1; epsilon=1, N=-2. Now, granted one could simply define N=1 for all epsilon>.25, to satisfy this particular definition; and as far as I can see that kind of trick could applied in all cases; so if the limit is defined in a version of the definition that does not include the restriction to positive N, then it would be defined in the version given above as well. However, if specifying that N is positive is unnecessary, then I don't see any reason for doing it.

This similar definition is given a couple of pages later in the same textbook:

Let f be defined on an open interval containing a, except possibly at a itself.

[math]\lim_{x\rightarrow a} f(x)=\infty[/math]

means that for every positive number M, there corresponds a [imath]\delta > 0[/imath] such that

if [imath]0 < \left | x-a \right | < \delta[/imath] then f(x)>M

{likewise, bolding added}

, and I see a similar trick for this as for the first one; but again why specify that M must be positive?

(Note, that in the textbook there are nearly identical definitions involving negative infinity, with the appropriate inequalities reversed and with "positive" replaced by "negative"; and likewise, I wonder the same thing about the negatives corresponding to the positives in the definitions that I have quoted, but I expect that the answer will be essentially identical.)

The other question that I have is that the "Let f be defined ..." sentence in the second definition is exactly identical to the first sentence in the definition given in this textbook for limits where both the limiting value and the approached x are finite. But is there some reason that there is no sentence at the start of the first definition requiring that f be defined for all x larger than some number a?