WEBVTT 00:00:04.000 --> 00:00:07.720 Okay, we're going to start off by talking about a bunch of limit laws. 00:00:07.980 --> 00:00:14.625 So today we're talking about, we're starting to figure out limits, not based on 00:00:14.625 --> 00:00:15.375 graphs, 00:00:15.280 --> 00:00:20.961 not based on tables or plugging in values, based on just given the function, the 00:00:20.961 --> 00:00:21.711 equation. 00:00:21.340 --> 00:00:23.120 Okay, let me go over some of these laws. 00:00:23.120 --> 00:00:28.120 You don't need to write much notes, but we'll refer back to these many times. 00:00:32.880 --> 00:00:35.920 Just let me give you a rough idea what it is. So the first one, 00:00:37.480 --> 00:00:39.700 the limit of the sum of two functions, 00:00:41.700 --> 00:00:47.689 we can find each limit separately. Limit of f plus the limit of g. So the limit of a 00:00:47.689 --> 00:00:49.580 sum is the sum of limits. 00:00:51.860 --> 00:00:57.820 The first five, this is only true if the actual limits exist. 00:00:58.380 --> 00:00:59.320 They equal some number. 00:01:00.880 --> 00:01:01.780 Equal some other number. 00:01:03.260 --> 00:01:07.720 If one of the two does not exist, the law does not apply. 00:01:09.400 --> 00:01:14.360 And then the limit of the difference is the difference of limits. 00:01:16.340 --> 00:01:22.989 Okay, any time you have a constant in front of a function, we can bring that outside 00:01:22.989 --> 00:01:23.739 the 00:01:23.380 --> 00:01:24.280 limit. 00:01:23.780 --> 00:01:33.316 So for example, limit as x goes to 5 or something of 7 times the square root of x 00:01:33.316 --> 00:01:34.320 plus 3. 00:01:34.500 --> 00:01:35.400 I'm just making some up. 00:01:35.360 --> 00:01:42.440 If we want to we can just bring this guy outside seven times whatever the limit is 00:01:50.140 --> 00:01:55.335 Number four the limit of a product of two different functions is the product of the 00:01:55.335 --> 00:01:56.085 limits 00:01:56.380 --> 00:01:59.320 For example the limit as x goes to zero 00:02:01.120 --> 00:02:04.500 square root of x times e to the x 00:02:05.760 --> 00:02:14.380 Anytime you have a product that's equal to the limit of the first function times the 00:02:14.380 --> 00:02:19.600 limit whatever the second function is. 00:02:20.360 --> 00:02:25.000 Again that's only true if this equals some number and this equals some number. 00:02:27.980 --> 00:02:32.120 The limit of the division of two functions is the division of the limits. 00:02:32.780 --> 00:02:40.403 This is only true if both of them equal some number and the bottom one does not 00:02:40.403 --> 00:02:41.300 equal zero. 00:02:53.260 --> 00:03:01.120 This one, if we have some function to any power, for example limit as x goes to... 00:03:09.280 --> 00:03:16.180 the fifth root of 4 to the power x squared. 00:03:17.840 --> 00:03:21.160 Well, that's not the best example. Let me go with cosine of x. 00:03:23.760 --> 00:03:29.220 The fifth root is what power? One fifth power. 00:03:29.220 --> 00:03:36.180 So that's the fifth root or one fifth power of the limit. 00:03:46.220 --> 00:03:51.580 Number seven, the limit of a constant is just equal to constant. 00:03:51.720 --> 00:03:52.620 Why is that? 00:03:53.800 --> 00:04:02.140 Alright, we have some constant function, no matter what x is. 00:04:03.180 --> 00:04:10.528 Whatever a is, as you approach a from the left and the right, it's always equal to 00:04:10.528 --> 00:04:11.278 whatever 00:04:10.960 --> 00:04:13.400 that constant is, it is whatever that number is. 00:04:17.000 --> 00:04:19.760 Don't worry about 8 for right now. 00:04:26.340 --> 00:04:28.900 Let's skip the next few. 00:04:29.780 --> 00:04:30.820 Let's turn to the back side. 00:04:33.820 --> 00:04:37.180 It says if f is a polynomial or rational function, 00:04:38.560 --> 00:04:40.280 and a is in the domain. 00:04:44.100 --> 00:04:47.040 All you have to do is directly substitute in that number 00:04:47.040 --> 00:04:47.940 into the function. 00:04:49.600 --> 00:04:51.540 We talked a little bit about this last Wednesday. 00:04:54.760 --> 00:04:55.840 Why can we do that? 00:05:01.760 --> 00:05:02.660 Yes. 00:05:03.520 --> 00:05:05.780 This is continuous everywhere. 00:05:06.560 --> 00:05:10.620 This is continuous at every number in its domain. 00:05:11.240 --> 00:05:12.860 Everything except what makes the denominator zero. 00:05:13.320 --> 00:05:16.300 So in fact, we're gonna use direct substitution, 00:05:16.800 --> 00:05:18.920 not only for polynomials and rational functions, 00:05:19.580 --> 00:05:23.360 any continuous function. 00:05:29.360 --> 00:05:31.440 Which are all the functions you know. 00:05:33.060 --> 00:05:37.580 polynomials, rational functions, exponential functions, logarithmic functions, 00:05:38.720 --> 00:05:45.153 all trigonometric functions. As long as that a value is in the domain, we just 00:05:45.153 --> 00:05:46.440 directly substitute it. 00:05:51.300 --> 00:05:56.020 And then we'll go over these last two some other day. 00:06:05.420 --> 00:06:06.320 Alright. 00:06:07.540 --> 00:06:10.420 Let's pull this tricks... 00:06:13.000 --> 00:06:14.400 for... 00:06:14.400 --> 00:06:15.420 and... 00:06:17.760 --> 00:06:19.980 We're gonna have a big list. 00:06:21.860 --> 00:06:25.581 Okay, so leave a lot of room and then maybe start working on the examples on the 00:06:25.581 --> 00:06:26.331 next 00:06:25.800 --> 00:06:26.700 page. 00:06:28.640 --> 00:06:31.940 Number one trick is direct substitution. 00:06:40.180 --> 00:06:43.400 This is almost always how we start, every limit. 00:06:44.760 --> 00:06:48.540 Ok, so we tried plugging in the value we are approaching. 00:07:15.280 --> 00:07:17.400 What type of function is this? 00:07:19.260 --> 00:07:20.160 Rational. 00:07:21.900 --> 00:07:25.240 We really don't even need to identify as rational for right now but 00:07:25.240 --> 00:07:29.320 we always just try plugging in the number 3 00:07:31.640 --> 00:07:33.400 I'd do it on the side somewhere 00:07:33.400 --> 00:07:38.300 3 squared minus 4 over 3 minus 2 00:07:38.300 --> 00:07:41.540 what is that, 9 minus 4 00:07:41.540 --> 00:07:51.780 is 5 over 1 is 5. If you end up with a number that is the limit. Okay done easy 00:07:51.780 --> 00:07:58.780 peasy. Okay why does this work? As long as this number is in the domain of a 00:07:58.780 --> 00:08:02.200 continuous function just directly substitute the limit from the left 00:08:02.200 --> 00:08:06.859 always leave the limit from right will equal the function value. Okay so that's the 00:08:06.859 --> 00:08:07.609 first trick. 00:08:13.780 --> 00:08:14.900 Next example. 00:08:27.500 --> 00:08:29.920 So this is the same function, but now we're approaching it. 00:08:32.260 --> 00:08:34.480 We're approaching as X goes to two. 00:08:37.480 --> 00:08:40.100 Again, on the side, we always try direct substitution. 00:08:40.900 --> 00:08:44.500 2 squared minus 4 over 2 minus 2. 00:08:46.660 --> 00:08:49.780 And we get 0 divided by 0. 00:08:50.960 --> 00:08:52.340 What is 0 divided by 0? 00:08:55.620 --> 00:08:56.520 That's fine. 00:08:56.940 --> 00:08:57.880 It's not a number. 00:08:58.760 --> 00:08:59.660 OK. 00:09:00.700 --> 00:09:02.280 However, this is very important. 00:09:04.140 --> 00:09:11.660 Any time direct substitution produces 0/0 in the context of a limit, 00:09:12.100 --> 00:09:14.440 this is called an indeterminate form. 00:09:26.400 --> 00:09:29.520 Okay, why is it called an indeterminate form? 00:09:29.520 --> 00:09:30.960 Let me take a guess. 00:09:33.760 --> 00:09:36.840 Because we cannot determine what the value of the limit is. 00:09:37.200 --> 00:09:39.260 The limit might exist, the limit might not exist. 00:09:40.380 --> 00:09:41.880 We cannot determine that yet. 00:09:42.720 --> 00:09:46.700 So any time you get 0/0, it is an indeterminate form. 00:09:50.020 --> 00:09:51.760 And it means we have to do more work. 00:09:52.040 --> 00:09:53.740 Anybody know the trick for this one? 00:09:54.350 --> 00:09:57.450 We try factoring. 00:10:09.310 --> 00:10:18.471 In fact, every time you get 0 over 0, that almost always means x minus this number 00:10:18.471 --> 00:10:19.221 is 00:10:19.010 --> 00:10:21.330 both a factor of the numerator and denominator. 00:10:23.830 --> 00:10:27.750 And once you factor it, we should see this common factor. 00:10:34.270 --> 00:10:36.150 We're going to simplify it. 00:10:38.910 --> 00:10:41.390 Oh, before I do that, let me ask you this. 00:10:54.610 --> 00:11:03.470 If you're just given this function, are we allowed to just simplify this function? 00:11:05.470 --> 00:11:07.730 Are they the same function? 00:11:09.010 --> 00:11:11.130 Is this function the same as this function? 00:11:16.490 --> 00:11:17.650 Definitely not. 00:11:17.650 --> 00:11:19.330 They're way different functions. 00:11:21.210 --> 00:11:24.470 Okay, well not way different. 00:11:27.650 --> 00:11:33.570 This function has a hole in it. This function does not have a hole. 00:11:34.810 --> 00:11:35.710 Okay. 00:11:36.870 --> 00:11:40.395 So if you're just given a rational function, you can't just simplify that changes 00:11:40.395 --> 00:11:42.410 the function. We've got rid of the hole. 00:11:43.030 --> 00:11:44.050 Totally different function. 00:11:44.810 --> 00:11:45.710 However, 00:11:47.650 --> 00:11:52.710 As long as x is not equal to 2, they are exactly equal. 00:11:54.150 --> 00:11:55.050 We agree with that? 00:11:55.650 --> 00:11:59.490 No matter what number you plug in, like we did 3 last time. 00:12:00.130 --> 00:12:03.550 3 minus 2 over 3 plus 2 over 3 minus 2. 00:12:04.390 --> 00:12:07.570 All these happen to be 1, but this number always simplifies to this number. 00:12:08.650 --> 00:12:09.970 As long as x is not 2. 00:12:10.530 --> 00:12:15.916 So these are the exact same functions, however, are everywhere except at the value 00:12:15.916 --> 00:12:17.070 of x equals 00:12:17.070 --> 00:12:21.110 2. So why can we simplify inside limits? 00:12:23.590 --> 00:12:29.730 Okay so we are going to simplify and say that this is the limit as x approaches 00:12:29.730 --> 00:12:37.930 2 of x plus 2. The reason we could do that inside of a limit is because when 00:12:37.930 --> 00:12:43.090 we're approaching 2 we don't care what happens at 2. All we care about is 00:12:43.090 --> 00:12:45.010 what's happening from the left and from the right. 00:12:47.450 --> 00:12:50.310 Since we're approaching 2, we're never equal to 2. 00:12:50.430 --> 00:12:52.490 So these are always exactly the same number. 00:12:52.730 --> 00:12:54.330 And we can simplify it. 00:12:54.530 --> 00:12:58.990 So inside limits, common factors, we can simplify it. 00:12:59.150 --> 00:13:02.830 And then we go back to our previous trick, which is what? 00:13:04.490 --> 00:13:05.390 Direct substitution. 00:13:05.990 --> 00:13:07.210 So we plug it in. 00:13:07.510 --> 00:13:09.090 And we get 2 plus 2. 00:13:09.530 --> 00:13:10.430 And the answer is 4. 00:13:17.730 --> 00:13:26.230 Okay, so trick number two is factor and simplify. 00:13:29.710 --> 00:13:34.590 Okay, and we often do this trick when you end up with zero over zero. 00:13:41.510 --> 00:13:44.010 What's the first step we always do? 00:13:46.710 --> 00:13:47.610 Direct substitution. 00:13:48.390 --> 00:13:50.310 And again, I do it on the side. 00:13:51.390 --> 00:13:53.510 Get 5 over 3 minus 3. 00:13:54.350 --> 00:13:56.833 And again, direct substitution, doesn't matter if we're going from the left to the 00:13:56.833 --> 00:13:57.583 right. 00:13:57.010 --> 00:14:01.750 And we get 5 over 0. 00:14:04.210 --> 00:14:05.490 Is this an indeterminate form? 00:14:08.890 --> 00:14:10.670 This is not an indeterminate form. 00:14:11.990 --> 00:14:12.890 OK? 00:14:12.790 --> 00:14:19.171 Any time you have any number other than 0, divided by 0, it's not an indeterminate 00:14:19.171 --> 00:14:19.921 form. 00:14:20.770 --> 00:14:21.910 Not indeterminate. 00:14:27.170 --> 00:14:33.410 Okay, whenever you get a number over zero, the answer is either going to be infinity 00:14:34.350 --> 00:14:35.290 or negative infinity. 00:14:36.750 --> 00:14:42.030 Okay, here this rational function has a vertical asymptote of three. 00:14:44.390 --> 00:14:49.998 Once we know it's a number over zero, we now just have to determine which one it is 00:14:49.998 --> 00:14:50.748 positive 00:14:50.310 --> 00:14:51.210 or negative infinity. 00:14:51.910 --> 00:14:53.170 And how do we figure that out? 00:14:56.330 --> 00:15:01.850 How do we know if it's going up or down or just plug 00:15:01.850 --> 00:15:06.470 in a number very close to 3, but from the left, which 00:15:06.470 --> 00:15:07.370 would be what? 00:15:07.190 --> 00:15:08.090 2.9. 00:15:08.630 --> 00:15:10.210 2.9 is fine. 00:15:11.010 --> 00:15:14.210 5 divided by 2.9 minus 3. 00:15:15.230 --> 00:15:16.910 And all we care about is the sign. 00:15:17.030 --> 00:15:18.390 Is it positive or negative? 00:15:19.910 --> 00:15:24.810 5 divided by negative 0.1 is this positive or negative? 00:15:26.290 --> 00:15:27.370 Definitely negative. 00:15:32.110 --> 00:15:35.610 So what does that tell us? The answer is negative infinity. 00:15:43.130 --> 00:15:48.950 So again, any time you direct substitution, you get a number over zero. 00:15:50.090 --> 00:15:52.150 The answer is going to be either positive or negative. 00:15:54.870 --> 00:15:56.190 Okay, sort of. 00:15:57.730 --> 00:15:59.550 That's when it's going from the left to the right. 00:16:00.630 --> 00:16:08.110 If it was this question, the limit as x goes to 3 of 5 over x minus 3. 00:16:11.750 --> 00:16:14.730 This is not from the left or from the right, it's from both. 00:16:15.650 --> 00:16:17.250 And now we have to check both sides. 00:16:17.910 --> 00:16:20.950 If they both go to negative infinity, the answer is going to be negative infinity. 00:16:21.430 --> 00:16:29.030 So, here we just have to check what is the limit as x goes to 3 from the right. 00:16:33.910 --> 00:16:36.830 Again you substitute and you get 5 over 0. 00:16:37.290 --> 00:16:39.130 So we just have to determine whether it is positive or negative infinity. 00:16:41.610 --> 00:16:45.070 And we just plug in a number to the right of 0. 00:16:46.090 --> 00:16:49.170 5 over 3.1 minus 3. 00:16:50.210 --> 00:16:52.510 5 over.1 positive. 00:16:55.670 --> 00:16:56.570 This is positive. 00:16:58.410 --> 00:17:00.850 This limit is equal to positive infinity. 00:17:04.610 --> 00:17:06.450 So what is the limit as x goes to 3? 00:17:10.610 --> 00:17:12.430 Does not exist. 00:17:12.430 --> 00:17:16.986 From the left it went to negative infinity, from the right it went to positive 00:17:16.986 --> 00:17:17.736 infinity, 00:17:17.490 --> 00:17:18.990 so this does not exist. 00:17:29.210 --> 00:17:32.490 So let's write down this trick. 00:17:32.490 --> 00:17:45.026 Not really a trick, but if you get by direct substitution a number divided by zero, 00:17:45.026 --> 00:17:45.810 non-zero 00:17:48.750 --> 00:18:01.360 number, the limit will either equal positive infinity or negative infinity, or does 00:18:01.360 --> 00:18:02.330 not 00:18:02.330 --> 00:18:07.170 exist. You gotta check both the left and the right. 00:18:11.290 --> 00:18:14.590 Zero over zero is an indeterminate form. Any nonzero number 00:18:14.590 --> 00:18:19.630 divided by zero is not indeterminate. The answer is either positive or 00:18:19.630 --> 00:18:22.830 negative infinity. Or from the left and the right they're opposite. 00:18:23.910 --> 00:18:24.810 Does not exist. 00:18:25.420 --> 00:18:27.740 What is the first thing we always try? 00:18:29.000 --> 00:18:29.900 Direct substitution. 00:18:31.260 --> 00:18:34.400 Square root of 9 minus 3, over 9 minus 9. 00:18:36.320 --> 00:18:38.080 Zero over zero. 00:18:38.760 --> 00:18:40.420 And what do we call that? 00:18:41.500 --> 00:18:42.400 Indeterminate form. 00:18:43.700 --> 00:18:44.600 Indeterminate form. 00:18:46.100 --> 00:18:48.160 So it means we have to do more work. 00:18:49.840 --> 00:18:50.740 Okay. 00:18:51.800 --> 00:18:53.260 Anybody know the trick for this one? 00:18:55.360 --> 00:18:56.760 Multiply by the conjugate. 00:18:57.240 --> 00:18:58.140 Okay. So. 00:18:57.920 --> 00:19:09.880 We are going to multiply by the conjugate of either 00:19:09.880 --> 00:19:12.500 the numerator or the denominator. 00:19:15.000 --> 00:19:15.940 When do we do this trick? 00:19:16.180 --> 00:19:18.260 Typically, you've got to have two different terms. 00:19:20.120 --> 00:19:21.020 OK. 00:19:20.680 --> 00:19:24.120 Both of them have two terms, but especially when 00:19:24.120 --> 00:19:25.240 it involves a square root. 00:19:26.620 --> 00:19:27.520 OK. 00:19:26.920 --> 00:19:32.320 So we're going to multiply the numerator and denominator by the conjugate of one of 00:19:32.320 --> 00:19:33.070 them. 00:19:33.140 --> 00:19:38.480 In this case, use the conjugate of the numerator: square root of x plus 3. 00:19:49.140 --> 00:19:51.940 the limit 00:19:51.940 --> 00:19:54.740 as x approaches 9 00:19:58.140 --> 00:19:59.040 of (square root of x minus 3) 00:20:00.700 --> 00:20:02.100 times (square root of x plus 3) 00:20:02.100 --> 00:20:07.220 over (x minus 9) 00:20:07.220 --> 00:20:09.260 times (square root of x plus 3). 00:20:10.520 --> 00:20:11.420 Okay. 00:20:12.260 --> 00:20:15.220 The other thing is, don't distribute the denominator. 00:20:15.540 --> 00:20:16.760 Just leave it the way it is. 00:20:17.900 --> 00:20:19.440 x minus 9 00:20:19.440 --> 00:20:22.060 times square root of x plus 3. 00:20:23.020 --> 00:20:25.160 We really only distribute the side that, 00:20:25.720 --> 00:20:28.540 or that, uh, the numerator and denominator that 00:20:28.540 --> 00:20:30.400 we've multi- 00:20:30.400 --> 00:20:32.440 uh, who we chose the 00:20:32.440 --> 00:20:33.580 conjugate of. 00:20:34.300 --> 00:20:36.340 And, when you multiply conjugates, 00:20:36.340 --> 00:20:39.600 the two middle terms always add to 0. 00:20:43.280 --> 00:20:47.100 So we get the limit as x approaches 9 00:20:47.100 --> 00:20:51.340 of (x minus 9) over (x minus 9)(square root of x plus 3). 00:20:57.260 --> 00:20:59.260 and now what? 00:20:59.300 --> 00:21:03.780 if you try plugging in 9 again 00:21:03.780 --> 00:21:06.120 in direct substitution you still get zero over zero. 00:21:07.140 --> 00:21:09.820 However, we go back from our previous trick. 00:21:10.260 --> 00:21:12.220 Most of the time you get zero over zero, 00:21:12.320 --> 00:21:13.360 we have a common factor. 00:21:15.060 --> 00:21:17.700 In this case, we have a common factor of X minus nine. 00:21:18.820 --> 00:21:19.740 We do simplify. 00:21:30.380 --> 00:21:32.580 We try direct substitution now. 00:21:33.780 --> 00:21:37.500 I get one sixth. 00:21:56.140 --> 00:21:58.180 Okay, so, 00:22:01.560 --> 00:22:04.020 Let's call it trick number four. 00:22:30.700 --> 00:22:34.220 So we multiply the numerator and denominator by the conjugate of either the 00:22:34.220 --> 00:22:37.100 numerator or denominator. 00:22:48.520 --> 00:22:49.860 When do we do this? 00:22:49.860 --> 00:22:58.127 That is typically when there is a square root and two terms in either the numerator 00:22:58.127 --> 00:22:59.160 or denominator. 00:23:05.040 --> 00:23:10.744 If we try direct substitution here, we get (3 plus 0) to the negative 1 power minus 00:23:10.744 --> 00:23:11.494 3 00:23:11.080 --> 00:23:12.400 to the negative 1 power, all divided by 0. 00:23:14.940 --> 00:23:16.580 We get 0 over 0. 00:23:20.880 --> 00:23:23.000 What are we going to do here? 00:23:30.400 --> 00:23:32.340 What does this mean to the negative one power? 00:23:35.100 --> 00:23:36.000 One over. 00:23:35.640 --> 00:23:40.240 one over so this is 00:23:46.620 --> 00:23:57.400 okay yes wait no no to the negative one power simply means one divided by 00:23:57.400 --> 00:24:07.000 whatever it is just the reciprocal whatever is inside that exponent okay so 00:24:07.000 --> 00:24:10.740 here we have fractions inside of fractions sometimes we call that complex 00:24:10.740 --> 00:24:16.180 fractions not like a complex number just fractions inside fractions so our trick 00:24:16.180 --> 00:24:22.080 is just to make one fraction we don't want fractions inside of fractions so we 00:24:22.320 --> 00:24:33.324 need a common denominator. Multiply one fraction by 3 plus x over 3 plus x, and the 00:24:33.324 --> 00:24:36.560 other by 3 over 3. 00:24:48.550 --> 00:25:00.550 3 minus (3 plus x), over 3 times (3 plus x), all divided by x. 00:25:03.310 --> 00:25:07.310 Now we have one fraction divided by another fraction. 00:25:08.850 --> 00:25:09.770 Let me put that there. 00:25:10.730 --> 00:25:11.630 So we take the numerator, 00:25:19.370 --> 00:25:23.590 which, after distributing the negative, becomes 3 minus 3 minus x. 00:25:24.510 --> 00:25:26.390 And that 3 minus 3 is 0. 00:25:27.610 --> 00:25:33.030 So we get negative x over 3 times (3 plus x). 00:25:33.810 --> 00:25:36.810 And we multiply by the reciprocal of the denominator. 00:25:37.330 --> 00:25:38.750 That's what division is. 00:25:46.550 --> 00:25:48.090 And then what happens? 00:25:50.670 --> 00:25:52.250 We have a common factor. 00:25:52.250 --> 00:25:55.290 So it simplifies. 00:26:11.450 --> 00:26:14.290 Now we directly substitute in. 00:26:30.330 --> 00:26:34.730 And when do I stop writing the word limit? 00:26:36.350 --> 00:26:39.910 We stop right when we actually find the limit or take the limit. 00:26:39.910 --> 00:26:46.390 so I did right here now I'm gonna directly substitute it and I got a 00:26:46.390 --> 00:26:52.090 number this one we don't write the word limit anymore as we say we took the 00:26:52.090 --> 00:26:57.370 limit the limit as it approaches is this everywhere else we got to be writing the 00:26:57.370 --> 00:27:05.577 word limit as X approaches whatever so there it is this is negative one ninth so 00:27:05.577 --> 00:27:06.327 let's 00:27:06.150 --> 00:27:09.150 Let's call this... 00:27:11.150 --> 00:27:12.450 Trick 5. 00:27:15.830 --> 00:27:16.730 Simplify... 00:27:19.110 --> 00:27:20.010 Complex... 00:27:21.470 --> 00:27:22.370 Fractions... 00:27:22.870 --> 00:27:25.150 Which are... 00:27:25.790 --> 00:27:27.170 Fractions within fractions.