Question about Definitions of Limits Involving Infinity

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fagricipni
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Question about Definitions of Limits Involving Infinity

Postby fagricipni » Fri Apr 22, 2011 6:08 am UTC

I'm quoting from my calculus textbook here:

[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?
Last edited by fagricipni on Fri Apr 22, 2011 7:14 am UTC, edited 1 time in total.

letterX
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Re: Question about Definitions of Limits Involving Infinity

Postby letterX » Fri Apr 22, 2011 6:34 am UTC

fagricipni wrote:Now, one question that I have is that is there any reason to specify that N must be a positive number.

Not really*.

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

Again, the definition works just as well if M can be negative or 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.)

You're correct. It will be.

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?

If we're being really technical, there should be. Though, usually for most of the functions that come up in Calculus, its entirely obvious whether a function is defined all the way out to infinity or not.


* Going to clarify my glib answer: You're entirely correct that the definition works exactly as well if M is allowed to be any number. However, the thing about the definition of limits is that if you've proved that for any epsilon, there's some M(epsilon) such that {the rest of the definition}, you can always let M be larger. So we might as well make M be positive, and an integer, because the positive integers have lots of nice properties like being able to do induction to them. And in general people just seem to like the natural numbers, so they stick with them when they get the chance.

fagricipni
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Re: Question about Definitions of Limits Involving Infinity

Postby fagricipni » Fri Apr 22, 2011 7:13 am UTC

letterX wrote:However, the thing about the definition of limits is that if you've proved that for any epsilon, there's some M(epsilon) such that {the rest of the definition}, you can always let M be larger.


Aw, heck; I did mess up in my first post (I'll go fix that); the first definition used N and then I used M in describing the behavior of the function I defined. Anyway, what you have said seems to be the analogue of the fact that in the delta-epsilon definition where both numbers are reals that one does not have to find the largest possible delta for any given epsilon but merely any possible delta for a given epsilon.

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Re: Question about Definitions of Limits Involving Infinity

Postby skeptical scientist » Fri Apr 22, 2011 7:22 am UTC

fagricipni wrote: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?

No, there isn't. A function should be defined for all x bigger than some number a if we're going to talk about the limit as x approaches infinity.

Often times people just leave off that requirement, and take it as understood that "for all x>N, |f(x)-L|<e" means that f(x) has to exist for x>N (since otherwise how can we say |f(x)-L|<e?)
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Yesila
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Re: Question about Definitions of Limits Involving Infinity

Postby Yesila » Fri Apr 22, 2011 3:39 pm UTC

Something to think about... What happens when the limiting behavior (as [imath]x\to\infty[/imath]) of a function is already being realized (that is for some small [imath]\varepsilon[/imath] we have [imath]|f(x)-L|<\varepsilon[/imath]) for negative (or really any "small" ) [imath]x[/imath]-values?

If we can keep our function arbitrarily close to L for any [imath]x>M[/imath] AND [imath]M[/imath] is not too big e.g. [imath]M[/imath] is negative) then what we have is a function that is constant at least when [imath]x>M[/imath]. That is we have a boring function. I already know what my function is going to do when x is big just by looking at the behavior it has for small x-values.

As others have said when your computing these differences ([imath]|f(x) -L|[/imath] it's "okay" to have negative M's ... but what that really means is either your function is borning, OR your epsilon is way to big. For any "interesting function" we expect that if the limit does exist, and we want our function values to be "very" close to that limit we'll need to use some "big" M values.

I suspect that the "requirement" to use a positive M is just a carry over from the question "what happens to [imath]f(x)[/imath] when we look at really big positive numbers." The answer of course is [imath]\lim_{x\to\infty} f(x)[/imath]. If it so happens that [imath]f[/imath] is already close to [imath]L[/imath] anytime [imath]x>M[/imath] for some negative [imath]M[/imath].... well fine...at least we have our answer for those "big" x-values that we really wanted to know about.

fagricipni
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Re: Question about Definitions of Limits Involving Infinity

Postby fagricipni » Fri Apr 22, 2011 8:07 pm UTC

@Yesila

In the case that I have given in which
[math]\lim_{x\rightarrow \infty}\frac{1}{x+3}[/math]

is being considered, then one advantage of not having the positive requirement is so that one can say that for a given [imath]\epsilon[/imath], that:
[math]N=\frac{1}{\epsilon}-3[/math]
without having to use a piecewise function to prevent N from being non-positive for a large enough [imath]\epsilon[/imath].

This simplification is the principle practical reason that I can see for not requiring that the numbers N and M to be positive in the definitions of limits involving infinity; I ,also, though, have a component of thinking that not requiring them to be positive if they don't need to be is an application of the principle of: "Things should be as simple as possible, but no simpler."


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