[MUSIC] There are subtleties even to things that appear as simple as addition.
Here, I've got some addition problems. 4279 + 1202,
4279 + 1190, 4269 + 1207,
42731202. + 1191,
and 4270 + 1100 You'll notice that they're all close to this problem.
The numbers that I've been listing off are all hovering around this problem.
Anyway, I'm going to give out these problems to some people and have them try
to do them. [MUSIC] Hi, I'm [UNKNOWN] and I'm Math
junior undergraduate at Ohio State. [MUSIC] Yeah. My name is Jacob Turner.
I'm a graduate TA here at OSU. [MUSIC] Alright.
People are finished doing the arithmetic problems.
Let's record the answers. So
here, 4270 + 1200 was 5470. 4279 + 1202 was 5481.
The next one is 5476, I've got 5467, and the last one was 5464.
what do all these numbers have in common? They're all really close together.
Is that just an accident? Of course, it's not an accident, right?
Here's the fact. Near the sum of two numbers is the sum of
two nearby numbers. These arithmetic problems are not just
random arithmetic problems. Look at the numbers I'm asking them to
add. 4277 and 1190,
those Those numbers are really close to 4273 and 1191.
Which is really close to 4270 and 1200. Which is really close to 4269 and 1207,
alright? Near the sum of two numbers is the sum of
two nearby numbers, all of the answers are nearby as well.
How does this relate to limits? Let's take a look.
Here's how it relates to limits. The limit of f of x plus g of x as x
approaches a, is the limit of f of x as x approaches a plus the limit of g of x, as
x approaches a. How is this related to those arithmetic
problems? Well, remember what this limit is saying.
This is saying, what can I make f of x plus g of x close to, if I'm willing to
make x sufficiently close to a. Well, it's going to be close to whatever
I can make f of x close to added to whatever I can make g of x close to,
right? It's the same kind of setup, right?
Near the sum of two values is the sum of the nearby values.
For the limit of a sum is the sum of the limits.
We can use this fact to do some calculations.
Let's see how. So, here's a limit problem.
The limit of x squared plus x as x approaches two.
I really want you to resist the temptation to just plug in two.
We're going to be using our limit laws to try to evaluate this limit.
Now, this is the limit of a sum, and the limit of the sum is the sum of the limits
provided the limits exist. So, this limit of x squared plus x is
equal to the limit of x squared as x approaches 2 plus the limit of x as x
approaches 2. Now, what's the limit of x squared?
Because the limit of the products is also the product of the limits provided the
limits exist, this is the limit of a product.
This is the limit of x times x. That's what x squared means, it's x times
x. So, I could rewrite this as the limit of
x times x, as x approaches 2+. Now what's the limit of x as x approaches
2? This is asking, what does x get close to
when x gets close to 2? Or, some more precisely, what can I
guarantee that x is close to if I'm willing to make x sufficiently close to
2? The limit of x as x approaches 2 is 2,
alright? So, this limit is just two.
Now, this a limit of a product, and the limit of a product is the product of the
limits provided the limits exist. So, this limit is the limit of x as x
approaches 2 times the limit of x as x approaches to +2.
Now again, the limit of x as x approaches two, right?
What can I guarantee that x is close to if x is close to 2?
Well, two. So, this is just 2.
This limit is the same thing, it's again just 2.
And here, I have +2. 2 * 2 + 2 is 6, which is the value of the
limit of x squared plus x as x approaches 2.
The takeaway message here is ask not what your country can do for you but what you
can do for your country. Or in other words, that the limit of a
sum is the sum of the limits provided the limits exist.
It's the same rhetorical device right, x-y, y-x.
Alright, the limit of a sum is the sum of the limits provided the limits exist.
I hope this is very memorable because these kinds of chiastic rules are going
to be used throughout our time together in order to evaluate limits.
Soon, we're going to see that the same sort of pattern holds not just for sums,
but for differences, for products and almost for quotients.
Good luck. [MUSIC]