In this video I want to talk about the definition of limit and what that means conceptually and
then go over how you can write a proof that a limit converges to some real number L using the
definition of limit. The definition of limit is something you'll learn in your first semester
course of real analysis and I didn't really understand it when they first went over it
but you'll do so many examples that it'll become second nature to you by the end of the course.
The definition of limit, if you want to prove that the limit Xn converges to the number L,
then the definition is that for any epsilon greater than 0 there exists an M in the
naturals such that when n is greater than or equal to M the absolute value of Xn minus L is
less than epsilon. That's a little bit of a weird definition but the idea of a limit is a little bit
weird itself. Sometimes you can have a sequence and none of the values of the sequence will ever
equal L but the limit will still be L if it goes arbitrarily close to L as n goes to infinity,
in this case. It never has to actually be L, it just has to be in the neighborhood. And if it's
outside of the neighborhood, there can only be finite--a finite many that are outside the
neighborhood. There can only be finite exceptions as the limit goes to infinity. That's covered in
this definition. If you just do an example, okay, the most simple example would just be a constant.
You know that the limit as n goes to infinity of 3 equals 3. If you have 3 minus the limit,
which is 3, less than epsilon. This would equal the absolute value of 0, which is just 0,
less than epsilon. By definition, epsilon has to be greater than 0 already. For any number,
that's going to be true. For any number, the limit of 3 will equal 3. This would be true for
any n greater than or equal to 1. But sometimes the first couple of values of n won't hold true
for this definition and so sometimes this number is greater than 1 as long as after an
arbitrary many it does approach L. Let's do a more complicated example next. This problem says, use
the definition of limit to prove that the limit as n goes to infinity of n cubed over 2n squared
plus n minus n over 2 equals 0. One of the good things about these definition of limit problems
is that they'll almost always give you what the limit is because they really just want to see
if you know how to apply the definition of limit. Again, it's kind of formulaic how you want to go
about solving or writing out the proof for these problems. Usually on a scratch piece of paper,
what the first thing you'll want to do is go ahead and write out what Xn minus L is in the absolute
value. You'll you should be able to simplify that and get to a point where it'll look like Xn minus
L is less than some number over n or something similar to that. And then you'll be able to apply
the idea of epsilon being greater than 0 and the rest of the definition. And then you rewrite it
using the definition first and this part will just make sense. If that didn't make any sense, let me
just go ahead and write out the proof. You're given that the limit equals 0, so by definition
you're going to have n cubed over 2n squared plus n minus n over 2 minus 0 in the absolute
value. Minus 0, already you know you can ignore that and let's go ahead and try to simplify this
into one fraction. We get n cubed times 2 minus n times 2n squared plus 9 all over 2n squared plus n
times 2. And that's going to be in the absolute value. When you simplify that, we get 2n cubed
minus 2n cubed minus 9n over 4n cubed plus 18. And that's absolute values. Not coincidentally,
you can cancel out the n cubes. The absolute value of 4n squared plus 18 is already positive,
so that's just going to be 4n squared plus 18. And the absolute value of 9n, let's just write that
out. And that's going to be 9n over 4n squared plus 18. We have simplified this part as much
as possible but if you see that 4n squared plus 18 is greater than 4n squared plus 0, then you
can show that 1 over 4n squared plus 18 has to be less than 1 over 4n squared. That means that
9n over 4n squared plus 18 has to be less than 9n over 4n squared. Now we've actually bounded
because this was our--this was from the definition of limit. This was Xn minus L. Now we've actually
been able to bound Xn minus L as less than 9n over 4n squared, which actually equals 9 over
4n. That's the number over n that I was talking about earlier. Now we can go ahead and use this
information and write out our proof. What we're going to say is, for any epsilon greater than 0,
9 over 4 epsilon is greater than 0 and 9 over 4 epsilon exists in R. By the Archimedean property,
we know that there exists an M the naturals such that M is greater than 9 over 4 epsilon. When we
have an n that's greater than or equal to M, we have n is greater than or equal to M, which
is greater than 9 over 4 epsilon, which means that 1 over n is less than or equal to 1 over M,
which is less than 1 over n over 4 epsilon. Then we know that Xn minus L equals 9n over 4n squared
plus 18. If you're counting this upper part as scratch paper, then you want to go ahead and write
that out. Once you get to that, we know that that is less than 9 over 4n and we know that so if you
multiply this by 9/4, that would give you (9/4)n is less than or equal to (9/4)M less than or equal
to 9/4 times 9 over 4e. That gives you less than or equal to (9/4)M which is less than or equal
to 9 over 4 (9/4) epsilon. That equals 9/4 times 4 epsilon over 9, which not coincidentally just
equals epsilon. Then you have that the absolute value of Xn minus L equals 9n 4n squared plus 18,
which is less than epsilon. You have that Xn minus L is less than epsilon. Thus by the
definition of limit, Xn, which equals--what was it? Nine cubed--9 cubed over 2n squared
plus 9 minus 9 over 2 converges to a real number and its limit is 0. And that's all we needed to
prove it. Hopefully--I think I wrote that kind of sloppily--but hopefully that made sense.
Basically, I got to the point where I would just memorize that once you simplify it, 9 over 4n,
that means that epsilon, you want to write out that epsilon is greater than 0 and 9 over 4
epsilon is greater than 0. Whatever this was, if that was for example 3 over 4n, then you want to
say in your definition, if epsilon is greater than 0 then 3 over 4 epsilon is greater than 0. And
also before I end the video something I noticed I did, obviously here we said that M is greater
than n over 4 epsilon. When I was multiplying, I made the mistake here. This is less than. This
should just be less than not less than or equal to and I carried that mistake over here as well.
This should just be less than and then the rest of the problem is right and hopefully it made sense.