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[MUSIC] There are subtleties even to 
things that appear as simple as addition. 

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Here, I've got some addition problems. 
4279 + 1202, 

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4279 + 1190, 
4269 + 1207, 

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42731202. 
+ 1191, 

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and 4270 + 1100 You'll notice that 
they're all close to this problem. 

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The numbers that I've been listing off 
are all hovering around this problem. 

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Anyway, I'm going to give out these 
problems to some people and have them try 

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to do them. 
[MUSIC] Hi, I'm [UNKNOWN] and I'm Math 

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junior undergraduate at Ohio State. 
[MUSIC] Yeah. My name is Jacob Turner. 

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I'm a graduate TA here at OSU. 
[MUSIC] Alright. 

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People are finished doing the arithmetic 
problems. 

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Let's record the answers. 
So 

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here, 4270 + 1200 was 5470. 
4279 + 1202 was 5481. 

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The next one is 5476, I've got 5467, 
and the last one was 5464. 

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what do all these numbers have in common? 
They're all really close together. 

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Is that just an accident? 
Of course, it's not an accident, right? 

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Here's the fact. 
Near the sum of two numbers is the sum of 

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two nearby numbers. 
These arithmetic problems are not just 

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random arithmetic problems. 
Look at the numbers I'm asking them to 

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add. 
4277 and 1190, 

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those Those numbers are really close to 
4273 and 1191. 

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Which is really close to 4270 and 1200. 
Which is really close to 4269 and 1207, 

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alright? 
Near the sum of two numbers is the sum of 

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two nearby numbers, 
all of the answers are nearby as well. 

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How does this relate to limits? 
Let's take a look. 

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Here's how it relates to limits. 
The limit of f of x plus g of x as x 

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approaches a, is the limit of f of x as x 
approaches a plus the limit of g of x, as 

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x approaches a. 
How is this related to those arithmetic 

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problems? 
Well, remember what this limit is saying. 

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This is saying, what can I make f of x 
plus g of x close to, if I'm willing to 

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make x sufficiently close to a. 
Well, it's going to be close to whatever 

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I can make f of x close to added to 
whatever I can make g of x close to, 

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right? 
It's the same kind of setup, right? 

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Near the sum of two values is the sum of 
the nearby values. 

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For the limit of a sum is the sum of the 
limits. 

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We can use this fact to do some 
calculations. 

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Let's see how. 
So, here's a limit problem. 

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The limit of x squared plus x as x 
approaches two. 

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I really want you to resist the 
temptation to just plug in two. 

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We're going to be using our limit laws to 
try to evaluate this limit. 

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Now, this is the limit of a sum, and the 
limit of the sum is the sum of the limits 

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provided the limits exist. 
So, this limit of x squared plus x is 

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equal to the limit of x squared as x 
approaches 2 plus the limit of x as x 

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approaches 2. 
Now, what's the limit of x squared? 

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Because the limit of the products is also 
the product of the limits provided the 

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limits exist, this is the limit of a 
product. 

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This is the limit of x times x. 
That's what x squared means, it's x times 

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x. 
So, I could rewrite this as the limit of 

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x times x, as x approaches 2+. 
Now what's the limit of x as x approaches 

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2? 
This is asking, what does x get close to 

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when x gets close to 2? 
Or, some more precisely, what can I 

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guarantee that x is close to if I'm 
willing to make x sufficiently close to 

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2? 
The limit of x as x approaches 2 is 2, 

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alright? 
So, this limit is just two. 

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Now, this a limit of a product, and the 
limit of a product is the product of the 

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limits provided the limits exist. 
So, this limit is the limit of x as x 

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approaches 2 times the limit of x as x 
approaches to +2. 

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Now again, the limit of x as x approaches 
two, right? 

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What can I guarantee that x is close to 
if x is close to 2? 

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Well, two. 
So, this is just 2. 

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This limit is the same thing, it's again 
just 2. 

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And here, I have +2. 
2 * 2 + 2 is 6, which is the value of the 

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limit of x squared plus x as x approaches 
2. 

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The takeaway message here is ask not what 
your country can do for you but what you 

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can do for your country. 
Or in other words, that the limit of a 

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sum is the sum of the limits provided the 
limits exist. 

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It's the same rhetorical device right, 
x-y, y-x. 

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Alright, the limit of a sum is the sum of 
the limits provided the limits exist. 

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I hope this is very memorable because 
these kinds of chiastic rules are going 

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to be used throughout our time together 
in order to evaluate limits. 

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Soon, we're going to see that the same 
sort of pattern holds not just for sums, 

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but for differences, for products and 
almost for quotients. 

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Good luck. 
[MUSIC]