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Vipul: Okay. So this talk is going to be about limit at infinity for functions on real numbers
and the concept of limits of sequences, how these definitions are essentially almost the
same thing and how they differ.
Okay. So let's begin by reviewing the definition of the limit as x approaches infinity of f(x).
Or rather what it means for that limit to be a number L. Well, what it means is that
for every epsilon greater than zero, so we first say for every neighborhood of L, small
neighborhood of L, given by radius epsilon there exists a neighborhood of infinity which
is specified by choosing some a such that that is
the interval (a,infinity) ...
... such that for all x in the interval from a to infinity. That is for all x within the
chosen neighborhood of infinity, the f(x) value is within the chosen neighborhood of
L. Okay?
If you want to think about it in terms of the game between the prover and the skeptic,
the prover is claiming that the limit as x approaches infinity of f(x) is L. The skeptic
begins by picking a neighborhood of L which is parameterized by its radius epsilon. The
prover picks the neighborhood of infinity which is parameterized
by its lower end a. Then the skeptic picks a value x between a and infinity. Then they
check whether absolute value f(x) minus L [symbolically: |f(x) - L|] is less than epsilon.
That is they check whether f(x) is in the chosen neighborhood of L (the neighborhood
chosen by the skeptic). If it is, then the prover wins. The prover has managed
to trap the function: for x large enough, the prover has managed to trap the function
within epsilon distance of L. If not, then the skeptic wins. The statement is true if
the prover has a winning the strategy for the game.
Now, there is a similar definition which one has for sequences. So, what's a sequence?
Well, it's just a function from the natural numbers. And, here, we're talking of sequences
of real numbers. So, it's a function from the naturals to the reals and we use the same
letter f for a good reason. Usually we write sequences with subscripts, a_n type of thing.
But I'm using it as a function just to highlight the similarities. So, limit as n approaches
infinity, n restricted to the natural numbers ... Usually if it's clear we're talking of
a sequence, we can remove this part [pointing to the n in N constraint specification] just
say limit n approaches infinity f(n), but since we want to be really clear here,
I have put this line. Okay?
So, this limit equals L means "for every epsilon greater than 0 ..." So, it starts in the same
way. The skeptic picks a neighborhood of L. Then the next line is a little different but
that's not really the crucial part. The skeptic is choosing epsilon. The prover picks n_0,
a natural number. Now, here the prover is picking a real number. Here the prover is
picking a natural number. That's not really the big issue. You could in fact change this
line to match. You could interchange these lines. It wouldn't affect either definition.
The next line is the really important one which is different. In here [pointing to real-sense
limit], the condition has to be valid for all x, for all real numbers x which are bigger
than the threshold which the prover has chosen. Here on the other hand [pointing to the sequence
limit] the condition has to be valid for all natural numbers which are bigger than the
threshold the prover has chosen. By the way, some of you may have seen the definition with
an equality sign here. It doesn't make a difference to the definition. It does affect what n_0
you can choose, it will go up or down by one, but that's not
really a big issue. The big issue, the big difference between these two definitions is
that in this definition you are insisting that the condition here is valid for all real
x. So, you are insisting or rather the game is forcing the prover to figure out how to
trap the function values for all real x. Whereas here, the game is only requiring the prover
to trap the function values for all large enough
natural numbers. So, here [real-sense limit] it's all large enough real numbers. Here [sequence
limit] it's all large enough natural numbers. Okay?
So, that's the only difference essentially. Now, you can see from the way we have written
this that this [real-sense limit] is much stronger. So, if you do have a function which
is defined on real so that both of these concepts can be discussed. If it were just a sequence
and there were no function to talk about then obviously, we can't even talk about this.
If there's a function defined on the reals or on all large enough reals, then we can
try taking both of these. The existence of this [pointing at the real-sense limit] and
[said "or", meant "and"] it's being equal to L as much stronger than this [the sequence
limit] equal to L. If this is equal to L then definitely this [the sequence limit] is equal
to L. Okay?
But maybe there are situations where this [the sequence limit] is equal to some number
but this thing [the real-sense limit] doesn't exist. So, I want to take one example here.
I have written down an example and we can talk a bit about that is this. So, here is
a function. f(x) = sin(pi x). This is sin (pi x) and the corresponding
function if you just restrict [it] to the natural numbers is just sin (pi n). Now, what
does sin (pi n) look like for a natural number n? In fact for any integer n? pi times
n is an integer multiple of pi. sin of integer multiples of pi is zero. Let's make a picture
of sin ...
It's oscillating. Right? Integer multiples of pi are precisely the ones where it's meeting
the axis. So, in fact we are concerned about the positive one because we are talking of
the sequence (natural number [inputs]). Okay? And so, if you are looking at this sequence,
all the terms here are zero. So, the limit is also zero. So, this limit [the sequence
limit] is zero.
Okay. What about this limit? Well, we have the picture again. Is it going anywhere? No.
It's oscillating between minus one and one [symbolically: oscillating in [-1,1]]. It's
not settling down to any number. It's not... You cannot trap it near any particular number
because it's all over the map between minus one and one. For the same reason that sin(1/x)
doesn't approach anything as x approaches zero, the same reason sin x or sin(pi x) doesn't
approach anything as x approaches infinity. So, the limit for the real thing, this does
not exist. So, this gives an example where the real thing [the real-sense limit] doesn't
exist and the sequence thing [sequence limit] does exist and so here is the overall summary.
If the real sense limit, that is this one [pointing to definition of
real sense limit] exists, [then] the sequence limit also exists and they're both equal.
On the other hand, you can have a situation with the real sense limit, the limit for the
function of reals doesn't exist but the sequence limit still exists like this set up.
Now, there is a little caveat that I want to add. If the real sense limit doesn't exist
as a finite number but it's say plus infinity then the sequence limit also has to be plus
infinity. If the real sense limit is minus infinity, then the sequence limit also has
to be minus infinity. So, this type of situation, where the real sense limit doesn't exist but
the sequence exists, well, will happen in kind of oscillatory type of situations. Where
the real sense you have an oscillating thing and in the sequence thing on the other hand
you somehow manage to pick a bunch of points where that oscillation doesn't create a problem.
Okay?
Now, why is this important? Well, it's important because in a lot of cases when you have to
calculate limits of sequences, you just calculate them by doing, essentially, just calculating
the limits of the function defining the sequence as a limit of a real valued function. Okay?
So, for instance if I ask you what is limit ...
Okay. I'll ask you what is limit [as] n approaches infinity of n^2(n + 1)/(n^3 + 1) or something
like that. Right? Some rational function. You just do this calculation as if you were
just doing a limit of a real function, function of real numbers, right? The answer you get
will be the correct one. If it's a finite number it will be the same finite number.
In this case it will just be one. But any rational function, if the answer is finite,
same answer for the sequence. If it is plus infinity, same answer for the sequence. If
it is minus infinity, same answer as for the sequence.
However, if the answer you get for the real-sense limit is oscillatory type of non existence,
then that's inconclusive as far as the sequence is concerned. You actually have to think about
the sequence case and figure out for yourself what happens to the limit. Okay? If might
in fact be the case that the sequence limit actually
does exist even though the real sense [limit] is oscillatory. Okay.