1 00:00:00,000 --> 00:00:03,251 [MUSIC]. A lot of calculus is about understanding 2 00:00:03,251 --> 00:00:08,493 the qualitative features of functions. Nobody really cares about f of 17. 3 00:00:08,493 --> 00:00:12,276 But you might care about f of a big number, qualitatively. 4 00:00:12,276 --> 00:00:16,921 We're going to try to make this concept of big number a bit more precise. 5 00:00:16,921 --> 00:00:21,300 Well, here's how we're going to get around talking about a big number. 6 00:00:21,300 --> 00:00:26,675 We're going to talk about the limit. The limit of f of x as x approaches 7 00:00:26,675 --> 00:00:30,259 infinity equals to l. Means that f of x is as close as you want 8 00:00:30,259 --> 00:00:32,980 it to be to l, provided x is large enough. 9 00:00:32,980 --> 00:00:38,167 So instead of talking about just plugging in a big number, I'm going to say F at 10 00:00:38,167 --> 00:00:43,623 some big number, so to speak, is equal to L if I can make the output of F close to 11 00:00:43,623 --> 00:00:48,474 L by plugging in big enough numbers. At this point we've seen a bunch of 12 00:00:48,474 --> 00:00:53,163 different definitions of limits. But they're all united by a common theme. 13 00:00:53,163 --> 00:00:58,292 What if someone had asked us to cook up a definition of the limit of F of X equals 14 00:00:58,292 --> 00:01:02,618 infinity as X goes to infinity? Could we come up with a definition for 15 00:01:02,618 --> 00:01:04,966 this? Yeah, absolutely we can, here we go. 16 00:01:04,966 --> 00:01:09,786 The limit of F of X as X approaches infinity equals infinity means that I can 17 00:01:09,786 --> 00:01:14,050 make F of X as large as you want it to be provided X is large enough. 18 00:01:14,050 --> 00:01:18,337 Consistently when we're talking about infinity in limits we're never actually 19 00:01:18,337 --> 00:01:22,350 talking about a specific value. We're just talking about a value which is 20 00:01:22,350 --> 00:01:27,320 as big as you want it to be. Let's go do an example at the blackboard. 21 00:01:27,320 --> 00:01:31,835 There is a question. With the limit, of 2x over x+1 as x 22 00:01:31,835 --> 00:01:36,861 approaches infinity. Before we dive into this analytically, 23 00:01:36,861 --> 00:01:42,144 lets get some numeric evidence. This is my function, again fx)= of 2x 24 00:01:42,144 --> 00:01:45,637 over x+1.1. I want to know qualitatively, what 25 00:01:45,637 --> 00:01:50,067 happens when I plug in big numbers in this function? 26 00:01:50,067 --> 00:01:51,260 So, lets say f100). of 100. 27 00:01:51,260 --> 00:02:00,066 Well, that's not too hard to figure out. 22 times 100 is 200 and 100 plus one is 28 00:02:00,066 --> 00:02:04,799 101. Now, 200 divided by 101 is pretty close 29 00:02:04,799 --> 00:02:07,754 to 2. And there's nothing too special about 30 00:02:07,754 --> 00:02:10,514 100, right? If I'd done this four million I would 31 00:02:10,514 --> 00:02:15,113 have gotten two million over a million in one, which would be even closer to 2. 32 00:02:15,113 --> 00:02:19,425 Numerically, it looks like this limit's two, but we just figured that out by 33 00:02:19,425 --> 00:02:23,220 plugging in some big numbers. I want a more rigorous argument, some 34 00:02:23,220 --> 00:02:25,980 analytic that is limit is actually equal to two. 35 00:02:25,980 --> 00:02:30,502 How I'm going to proceed? My first guess would be to use the limit 36 00:02:30,502 --> 00:02:34,075 law for quotients. This is the limit of a quotient which is 37 00:02:34,075 --> 00:02:38,314 the quotient of limits provided the limits exist and the limit of the 38 00:02:38,314 --> 00:02:42,190 denominator is none zero. Bad news here is the limits don't exist. 39 00:02:42,190 --> 00:02:45,824 The limit of the numerator is infinity which isn't a number. 40 00:02:45,824 --> 00:02:48,670 So I can't use my limit law for quotients here. 41 00:02:48,670 --> 00:02:52,970 Instead I'm going to sneak up on this limit problem by wearing a disguise. 42 00:02:52,970 --> 00:02:55,937 I'm going to multiply by a disguised version of one. 43 00:02:55,937 --> 00:02:59,268 I'm going to multiply by one over x divided by one over x. 44 00:02:59,268 --> 00:03:03,282 Now this is just one. Admittedly, I have changed the function. 45 00:03:03,282 --> 00:03:06,306 The function's not defined anymore at zero. 46 00:03:06,306 --> 00:03:10,174 But for large values of X, this doesn't affect anything. 47 00:03:10,174 --> 00:03:13,479 And I'm taking the limit as X goes to infinity. 48 00:03:13,479 --> 00:03:17,136 So I only care about agreement at large values of X. 49 00:03:17,136 --> 00:03:21,496 I'll do some algebra. This limit has now the limit of 2X times 50 00:03:21,496 --> 00:03:25,012 one over X. Which is two divided by the limit of X 51 00:03:25,012 --> 00:03:29,340 plus one times one over X. Which is one plus one over X. 52 00:03:29,340 --> 00:03:34,252 Maybe it doesn't look like I've made a lot of progress here, but this is a huge 53 00:03:34,252 --> 00:03:37,050 progress. The limit of the numerator is now a 54 00:03:37,050 --> 00:03:40,594 number, it's two. And the limit of the denominator is also 55 00:03:40,594 --> 00:03:43,020 a number. And a non zero number, at that. 56 00:03:43,020 --> 00:03:48,056 So I can use my limit law for quotients. This is the limit of a quotient, which is 57 00:03:48,056 --> 00:03:51,787 the quotient of a limit. It's the limit of two, the numerator 58 00:03:51,787 --> 00:03:55,270 divided by the limit of the denominator, as x approaches infinity. 59 00:03:55,270 --> 00:03:59,954 The limit of the numerator is just the limit of a constant, which is 2. 60 00:03:59,954 --> 00:04:05,160 The limit of the denominator is a limit of a sum, which is the sum of the limits 61 00:04:05,160 --> 00:04:09,520 provided the limits exist, and they do. The limit of 1 is just 1. 62 00:04:09,520 --> 00:04:14,948 And what's the limit of 1 over x as x approaches infinity? 63 00:04:14,948 --> 00:04:21,195 Well that's asking, what is 1 over x close to when x is very large? 64 00:04:21,195 --> 00:04:29,184 Well I can make 1 over x as close to zero as I like if I'm willing to make x large 65 00:04:29,184 --> 00:04:33,588 enough. So the limit of 1 over x as x approaches 66 00:04:33,588 --> 00:04:39,938 infinity is zero. That means my original limit is 2 over 1 67 00:04:39,938 --> 00:04:43,114 plus zero, which is 2. [MUSIC].