1
00:00:00,000 --> 00:00:05,007
Here is an arithmetic problem, 
660 * 310. 

2
00:00:05,007 --> 00:00:13,318
I'm going give this exercise to Bart. 
So, the product of 660 and 310 is 

3
00:00:13,318 --> 00:00:18,113
204,600. 
But, what if I perturb these inputs a 

4
00:00:18,113 --> 00:00:23,654
little bit? 
Instead of assigning this multiplication 

5
00:00:23,654 --> 00:00:30,900
exercise, I could have assigned this 
multiplication exercise, 

6
00:00:30,900 --> 00:00:35,488
664 * 311. 
Let's give this to somebody else. 

7
00:00:35,488 --> 00:00:39,998
Hi. 
My name is Vadranna. 

8
00:00:39,998 --> 00:00:52,255
So, 664 * 311 is 206,504. 
Which isn't so far off of the answer that 

9
00:00:52,255 --> 00:00:57,137
Bart got when he multiplied 660 times 310 
and got 204,600. 

10
00:00:57,137 --> 00:01:00,750
So, look at this. 
We've got two different problems. 

11
00:01:00,750 --> 00:01:06,805
The input to these multiplication 
problems are similar, 

12
00:01:06,805 --> 00:01:13,521
the outputs are also similar. 
Let's do some more, very similar 

13
00:01:13,521 --> 00:01:20,346
multiplication problems. 
[SOUND] Hello, my name is Sean Gory. 

14
00:01:20,346 --> 00:01:23,980
[SOUND] Oh, this is great. 
Look. 

15
00:01:23,980 --> 00:01:32,480
204,600, 206,504, 206,926, 204,702. 
We multiplied all of these pairs of 

16
00:01:32,480 --> 00:01:41,230
nearby numbers, and the result of the 
multiplications were also nearby. 

17
00:01:41,230 --> 00:01:45,722
There's a limit lesson hiding at all of 
this. 

18
00:01:45,722 --> 00:01:53,210
If the limit of f of x as x approaches a 
is L, and the limit of g of x as x 

19
00:01:53,210 --> 00:02:01,497
approaches a is M, then the limit of f of 
x times g of x as x approaches a is equal 

20
00:02:01,497 --> 00:02:07,687
to L times M. 
In other words, the limit of a product is 

21
00:02:07,687 --> 00:02:13,877
the product of the limits provided those 
limits exist [SOUND]