WEBVTT 00:00:05.900 --> 00:00:21.900 All right. So what's the first thing we always try? Direct substitution. Over here, 00:00:21.900 --> 00:00:35.880 we get 1 over (0 times the square root of 1 plus 0), minus 1 over 0. 00:00:35.880 --> 00:00:45.913 0 go to? Well, in limits, it will either go to positive or negative infinity. It 00:00:45.913 --> 00:00:46.663 depends 00:00:46.540 --> 00:01:06.827 on the left to the right. So let's not worry about the left and right phenomenon. So 00:01:06.827 --> 00:01:08.020 this 00:01:08.020 --> 00:01:11.678 one over zero goes to either positive or negative infinity. This goes to either 00:01:11.678 --> 00:01:12.428 positive or 00:01:12.200 --> 00:01:20.960 negative infinity okay and we get this what is infinity minus infinity not zero 00:01:22.000 --> 00:01:36.340 we don't know what it is it's another indeterminate form okay if these both go 00:01:36.340 --> 00:01:40.540 to positive infinity or sorry let's say the one on the left goes to positive and 00:01:40.540 --> 00:01:41.440 this goes to negative. 00:01:42.080 --> 00:01:44.780 This would be infinity plus infinity, 00:01:44.860 --> 00:01:46.540 which does go to infinity. 00:01:47.740 --> 00:01:48.640 But we don't know, 00:01:48.540 --> 00:01:50.020 because we didn't do it from the left and the right. 00:01:53.160 --> 00:01:55.800 So anytime you get infinity minus infinity, 00:01:57.160 --> 00:01:58.800 indeterminate, we gotta do some more work. 00:02:01.180 --> 00:02:02.800 All right, so this next trick, 00:02:05.460 --> 00:02:08.280 is we just need to combine them into one fraction. 00:02:08.860 --> 00:02:11.400 So we get a fraction minus another fraction. 00:02:11.460 --> 00:02:13.420 We combine them into one fraction. 00:02:13.780 --> 00:02:16.520 So we just need to get a common denominator 00:02:17.560 --> 00:02:19.280 of the two fractions. 00:02:27.200 --> 00:02:28.100 So we just need to multiply this fraction by something. What do we multiply it by? 00:02:34.960 --> 00:02:37.620 So we just need to multiply this guy by something. 00:02:37.620 --> 00:02:39.260 What do we multiply this guy by? 00:02:41.340 --> 00:02:45.580 I have 1 plus x over 1 plus x. 00:02:46.160 --> 00:02:47.880 Now they have a common denominator. 00:02:50.820 --> 00:02:59.082 And now if we try direct substitution, we get 1 minus the square root of 1 plus 0, 00:02:59.082 --> 00:03:00.000 over 0. 00:03:03.720 --> 00:03:06.220 Now we get 0 over 0, which is? 00:03:07.100 --> 00:03:09.320 A different indeterminate form. 00:03:09.960 --> 00:03:22.680 we're gonna do some more work so now what so we go back to one of our other 00:03:22.680 --> 00:03:28.540 tricks multiplied by the conjugate because we got two terms here especially 00:03:28.540 --> 00:03:50.281 one involving square root. So let's multiply by 1 plus 1 plus x. And again, the 00:03:50.281 --> 00:03:51.640 bottom 00:03:51.640 --> 00:03:55.653 or the denominator, which we did not choose the numerator, don't multiply that out. 00:03:55.653 --> 00:03:56.403 Just 00:03:55.940 --> 00:04:09.361 keep that all factored. Yeah. Again the two middle terms of the conjugate add to be 00:04:09.361 --> 00:04:10.200 zero. 00:04:10.200 --> 00:04:18.200 So we get 1 plus 0 minus 1 plus x. 00:04:21.140 --> 00:04:23.740 Now we distribute this guy. 00:04:26.780 --> 00:04:28.640 And now what? 00:04:32.820 --> 00:04:37.340 one of our other tricks. 00:04:42.000 --> 00:04:45.240 cancel the common factor x. 00:04:48.640 --> 00:04:51.280 And now we return to direct substitution and see what happens. 00:04:53.940 --> 00:04:59.480 The denominator becomes 1 times 1 plus 1, so the limit is negative one-half. 00:05:01.980 --> 00:05:06.660 Okay, so. 00:05:06.660 --> 00:05:12.340 So the very first trick was to combine multiple fractions or the sum or 00:05:12.340 --> 00:05:14.180 difference of fractions into one fraction. 00:05:17.320 --> 00:05:20.480 And then also the other trick is keep trying more tricks. 00:05:30.560 --> 00:05:32.340 Another technique. 00:05:45.360 --> 00:05:46.760 Alright. 00:05:46.760 --> 00:05:47.660 Alright. 00:05:50.580 --> 00:05:56.020 Direct substitution: what is sine of zero? 00:06:01.480 --> 00:06:17.100 Oh what do we do here okay this next 00:06:17.100 --> 00:06:33.760 trick very important one you just memorize the answer memorize okay this 00:06:33.760 --> 00:06:38.500 is one of the ones we're just gonna memorize the answer is one here it's not 00:06:38.500 --> 00:06:39.400 obvious 00:06:42.480 --> 00:06:45.760 Ok, so here is our function sine of x and x. 00:06:47.960 --> 00:06:53.735 As you approach zero, these two functions get closer and closer and closer to each 00:06:53.735 --> 00:06:54.485 other. 00:06:54.120 --> 00:06:57.320 And when you divide two numbers that are very, very close, 00:06:58.360 --> 00:07:00.380 it gets closer and closer to 1. 00:07:04.480 --> 00:07:09.540 If I zoom in here, the closer you get to 0, the closer 00:07:09.540 --> 00:07:11.380 you can tell the difference between the two. 00:07:13.300 --> 00:07:18.720 And then if I show you this function, y equals sine x. 00:07:31.940 --> 00:07:33.280 It's the green function. 00:07:34.840 --> 00:07:37.080 There is a hole right there at x equals zero. 00:07:37.260 --> 00:07:39.900 That function doesn't exist at x equals zero. 00:07:41.440 --> 00:07:44.780 However, you approach from the left and the right, it gets closer and closer to one. 00:07:45.840 --> 00:07:49.000 So the trick is you just memorize this one. 00:07:50.360 --> 00:07:51.260 Okay. 00:07:53.560 --> 00:07:57.260 Okay. One of the limits we memorize is this one: sine of x over x. 00:07:59.240 --> 00:08:00.320 Sine of x over x. 00:08:00.320 --> 00:08:02.320 O-Rex... 00:08:04.880 --> 00:08:08.040 What about this? 00:08:10.160 --> 00:08:12.400 Is this also going to equal one or no? 00:08:13.900 --> 00:08:14.800 It's going to be negative. 00:08:16.980 --> 00:08:17.880 That's right. 00:08:17.860 --> 00:08:20.380 It's just a guess. That's not a bad guess. 00:08:21.260 --> 00:08:23.920 Not quite true, but it's not a bad guess. 00:08:25.640 --> 00:08:28.480 Is this going to equal one? Why or why not? 00:08:31.620 --> 00:08:34.680 Is this true? 00:08:34.680 --> 00:08:46.960 wrote down here yes okay right if you had this you take the numerator it's one 00:08:46.960 --> 00:08:54.480 multiplied by the reciprocal and you get exactly that okay so this limit is the 00:08:54.580 --> 00:09:08.800 limit as x approaches zero of one divided by sine of x over x. And one of the limit 00:09:08.800 --> 00:09:21.112 laws that I handed out is the limit of a quotient is the quotient of limits. So we 00:09:21.112 --> 00:09:22.480 could write 00:09:22.480 --> 00:09:30.760 this as the limit as X approaches 0 of 1 divided by the limit as X approaches 0 00:09:31.840 --> 00:09:48.120 sine of X over X and what is this limit? Limit of a constant is just that constant 00:09:48.840 --> 00:09:54.340 And this limit we just memorized is 1 is definitely equal to 1. 00:09:58.560 --> 00:10:05.300 Long story short, limit as x approaches 0 of sine of x over x or x over sine of x is 00:10:05.300 --> 00:10:06.200 1. 00:10:09.580 --> 00:10:10.600 Very important. 00:10:14.080 --> 00:10:15.780 So if we try direct substitution, 00:10:15.960 --> 00:10:20.280 we get 1 minus cosine of 0 over 0. 00:10:21.860 --> 00:10:23.420 What is cosine of 0? 00:10:25.980 --> 00:10:27.300 0 over 0. 00:10:37.420 --> 00:10:39.300 So, is this also going to equal one? 00:10:45.600 --> 00:10:47.560 It's going to be one? 00:10:48.500 --> 00:10:49.820 Some people shake their head no? 00:10:52.520 --> 00:10:56.940 We are also going to memorize this one, but let's figure it out before we get the 00:10:56.940 --> 00:10:57.690 answer. 00:11:01.100 --> 00:11:02.800 Here's a little trick. 00:11:04.800 --> 00:11:09.400 We're going to multiply by 1 plus cosine of x. 00:11:12.540 --> 00:11:18.160 which is just a conjugate of the numerator there. 00:11:21.520 --> 00:11:28.200 These add to be zero. 00:11:31.080 --> 00:11:32.880 Does that help anything? 00:11:34.500 --> 00:11:37.320 You direct substitution, you still get 0 over 0. 00:11:39.380 --> 00:11:46.160 However, I could substitute this for something. 00:11:46.320 --> 00:11:48.380 What is 1 minus cosine squared always equal to? 00:11:49.300 --> 00:11:50.200 Sine squared, right? 00:11:50.480 --> 00:11:55.000 Because sine squared plus cosine squared equals 1. 00:11:58.100 --> 00:12:01.000 So sine squared is one minus cosine squared. 00:12:15.640 --> 00:12:18.900 If you try direct substitution, we still get zero over zero. 00:12:20.800 --> 00:12:23.240 Any ideas what we could do from here? 00:12:26.640 --> 00:12:27.600 What's that? 00:12:37.740 --> 00:12:39.780 I'm not sure we could do that. 00:12:43.540 --> 00:12:47.760 However, sine squared is sine times sine, correct? 00:12:49.120 --> 00:12:52.920 We have an x down here, so we're going to rewrite this as this. 00:12:55.840 --> 00:13:09.500 sine of x over x times sine of x over 1 plus cosine of x. 00:13:17.060 --> 00:13:19.480 And what is the limit of this guy, guys? 00:13:22.260 --> 00:13:26.380 So the limit of a product is the product of limits. 00:13:27.360 --> 00:13:28.960 So I'm going to rewrite like this. 00:13:32.280 --> 00:13:33.680 this 00:13:33.680 --> 00:13:46.000 limit is equal to 1 and now we directly substitute and we get sine of 0 which is 00:13:46.000 --> 00:14:03.080 0 1 plus cosine of 0 which is 1 which is 0 over 1 plus 1 which the whole thing is 00:14:03.080 --> 00:14:23.105 simply zero. Okay, and what we typically do, or what I typically do, instead of 00:14:23.105 --> 00:14:24.440 writing 00:14:24.440 --> 00:14:27.725 the limit of the quotient, a limit times a limit, this is technically what's 00:14:27.725 --> 00:14:28.475 happening. 00:14:27.960 --> 00:14:38.989 what we typically do is just we say this thing goes to 1 and the limit x goes to 0 00:14:38.989 --> 00:14:39.739 of 00:14:39.540 --> 00:14:47.540 sine of x over 1 plus cosine of x and then we get 0 over 1 or 2 so if you ever 00:14:47.540 --> 00:14:53.900 see me like put arrows here that's just meaning whatever this function is it 00:14:53.900 --> 00:14:58.800 goes to whatever the limit of it goes to that number just kind of saves you time 00:14:58.800 --> 00:15:02.980 from writing the limit times limits on okay so this is another one we memorize 00:15:06.980 --> 00:15:10.660 this is really the only other one we memorize 00:15:13.900 --> 00:15:15.440 This is just slightly different. 00:15:16.020 --> 00:15:17.280 How is this different than 3 to the 1? 00:15:23.740 --> 00:15:29.500 The last one was 1 minus cosine of x, this is cosine of x minus 1. 00:15:31.880 --> 00:15:33.680 Is this also going to be equal to 0? 00:15:44.260 --> 00:15:48.240 Take a guess yes or no everybody give me a thumb. 00:15:53.220 --> 00:15:58.260 Okay let's work it out it does equal to one I'm sorry does equal to zero. 00:16:01.320 --> 00:16:08.020 If I factor out a negative out of that, I get the limit as x goes to zero. 00:16:11.280 --> 00:16:15.707 And then one of our limit laws if we have a coefficient in front what can I do with 00:16:15.707 --> 00:16:16.457 that? 00:16:18.440 --> 00:16:22.760 You could distribute what are my limit laws here? 00:16:28.940 --> 00:16:33.740 Or wasn't at the very top yeah, here it is 00:16:37.940 --> 00:16:39.360 This one right here. 00:16:40.060 --> 00:16:42.540 You have a coefficient in front, what can I do with the coefficient? 00:16:44.600 --> 00:16:47.580 We can bring it outside the limit right here. 00:16:55.320 --> 00:16:56.940 So this can come out. 00:16:56.940 --> 00:17:09.820 we get negative 1 times this limit and then let me rewrite these two 00:17:10.380 --> 00:17:15.980 negative cosine plus x plus 1 is 1 minus cosine of x 00:17:20.860 --> 00:17:22.380 and what's this equal to 00:17:25.100 --> 00:17:27.120 Zero, one of the ones we memorized. 00:17:27.480 --> 00:17:30.960 We just did, so it's negative one times zero, it is zero. 00:17:45.500 --> 00:17:50.220 You don't really have to memorize it, but again, the reciprocal of this is the same 00:17:50.220 --> 00:17:57.240 value and if these are swapped it's the same value. They're both 0, they're both 1 00:18:00.980 --> 00:18:06.100 Let me just write that down anyways. 00:18:08.280 --> 00:18:11.140 x over sin x 00:18:13.200 --> 00:18:14.540 also equals to 00:18:14.540 --> 00:18:16.500 one minute 00:18:19.220 --> 00:18:21.800 So the reciprocal has the same value here. 00:18:22.340 --> 00:18:23.580 And we can swap that. 00:18:23.580 --> 00:18:24.600 I'm going to do some algebra. 00:18:39.440 --> 00:18:41.020 All right, next one. 00:18:43.800 --> 00:18:47.220 I'll try direct substitution. 00:18:47.400 --> 00:18:54.560 We get 0 times secant of 0 times cosecant of 0. 00:18:57.140 --> 00:18:59.400 What is secant of zero? 00:18:59.800 --> 00:19:04.100 What is cosine of zero? 00:19:07.160 --> 00:19:08.800 Which is one. 00:19:10.940 --> 00:19:12.140 And cosecant of x? 00:19:14.540 --> 00:19:15.540 What is sine of zero? 00:19:17.660 --> 00:19:20.100 Zero. So one of this does not exist. 00:19:25.160 --> 00:19:26.520 So, we got to do something else. 00:19:27.220 --> 00:19:28.220 What are we going to do? 00:19:31.360 --> 00:19:32.260 Any ideas? 00:19:36.320 --> 00:19:40.100 Figuring out secant, cosecant, cotangent, they are hard, right? 00:19:40.460 --> 00:19:42.560 Because then it is just the reciprocal sign goes in. 00:19:42.820 --> 00:19:45.400 So the trick is just change everything in terms of sine and cosine. 00:19:45.400 --> 00:19:53.180 So all trig functions can be changed in terms of sine and cosine. So this is the 00:19:53.180 --> 00:20:05.740 limit as x goes to 0, x times 1 over cosine of x times 1 over sine of x. 00:20:11.000 --> 00:20:12.860 Now what? 00:20:15.580 --> 00:20:18.340 What does this guy go to as x goes to zero? 00:20:19.760 --> 00:20:20.660 One. 00:20:20.380 --> 00:20:21.280 One. 00:20:21.420 --> 00:20:23.700 So if you want to split it up we can. 00:20:28.380 --> 00:20:36.653 The limit as x goes to zero of one over cosine times the limit as x goes to zero of 00:20:36.653 --> 00:20:37.480 x over 00:20:37.480 --> 00:20:38.680 sine of x. 00:20:41.380 --> 00:20:44.160 And then this direct substitution, we get 1. 00:20:46.020 --> 00:20:50.200 And this, we memorized as 1, the answer is 1. 00:20:54.580 --> 00:20:57.340 So the trick is just change all trig functions 00:20:57.340 --> 00:20:58.760 in terms of sine and cosine. 00:21:01.700 --> 00:21:09.440 So here it's not sine of x over x, sine of x over 7x. 00:21:11.400 --> 00:21:13.020 So what can we do with that? 00:21:17.820 --> 00:21:19.420 Yeah, all we do is we factor this is, 00:21:20.960 --> 00:21:23.540 it's just a coefficient of like 1 7th. 00:21:23.540 --> 00:21:29.140 And we factor it out. 00:21:38.700 --> 00:21:41.840 We have 1 7 times 1 is 1 7. 00:21:56.820 --> 00:21:59.260 That one shouldn't have been too hard, but let's try this one. 00:22:01.900 --> 00:22:04.840 Can I factor out a 5 here? 00:22:08.600 --> 00:22:09.500 Definitely not. 00:22:10.660 --> 00:22:12.780 You can't factor a number out from inside a trigonometric function. 00:22:14.220 --> 00:22:15.220 Uh oh, so now what? 00:22:17.900 --> 00:22:19.640 Multiply by 5 over 5. 00:22:20.560 --> 00:22:22.300 Okay, I like that. 00:22:24.560 --> 00:22:27.420 Multiply this by 5 over 5. 00:22:29.380 --> 00:22:33.580 And take this guy out 00:22:47.380 --> 00:22:49.240 Why did you want that 00:22:53.920 --> 00:22:54.820 It functions. 00:22:56.640 --> 00:23:00.300 Yeah, so even though it's not sine of x over x, 00:23:01.540 --> 00:23:05.180 it's the same exact thing. 00:23:06.900 --> 00:23:11.460 And as x goes to 0, 5x also goes to 0. 00:23:13.100 --> 00:23:16.440 And this goes to 0 at exactly the same rate. 00:23:17.120 --> 00:23:28.960 If you wanted to, you could let like, uh, u, uh, yeah, let me go, u is five X. 00:23:32.920 --> 00:23:38.400 And the limit as X goes to zero of u. 00:23:40.300 --> 00:23:41.200 Okay. 00:23:40.940 --> 00:23:44.440 This also goes to zero and you end up with something like this. 00:23:44.440 --> 00:23:52.140 you don't really have to do this, but sine of u over u. 00:23:52.900 --> 00:23:57.280 And this limit, whatever the variable is, is equal to 1. 00:24:03.420 --> 00:24:07.980 So the key here, as long as whatever is inside the sine 00:24:08.560 --> 00:24:11.800 has to be that same thing down here. 00:24:13.540 --> 00:24:16.960 This will equal to, well, in this case, 5. 00:24:18.180 --> 00:24:24.380 This will equal, in this case, 5. 00:24:24.380 --> 00:24:25.300 bring the letters out. 00:24:29.140 --> 00:24:33.720 Oh, here's a good one. 00:24:34.000 --> 00:24:34.900 All right. 00:24:34.380 --> 00:24:35.280 New topic. 00:24:35.140 --> 00:24:36.040 Yeah. 00:24:38.520 --> 00:24:46.640 We try direct substitution, we get 3 minus 3, or 3 minus 3, we get 0 over 0. 00:24:48.080 --> 00:24:49.980 What in the world are we going to do here? 00:24:58.640 --> 00:25:06.580 Uh, this absolute value, what kind of function is that? 00:25:08.460 --> 00:25:15.398 What's that? It's a piecewise function. Okay, let's look at the definition of the 00:25:15.398 --> 00:25:17.380 absolute value of x 00:25:17.380 --> 00:25:18.792 minus three. This is a piecewise function. It's either going to be x minus three or 00:25:18.792 --> 00:25:19.542 it's 00:25:18.880 --> 00:25:26.000 x minus three if x minus three is greater than or equal to zero and if x 00:25:26.000 --> 00:25:41.700 minus three is less than zero so x minus three it's either x minus three if x is 00:25:41.700 --> 00:25:43.500 Yeah, greater than or equal to zero. 00:25:44.620 --> 00:25:46.680 Sorry, greater than or equal to three. 00:25:49.320 --> 00:25:51.060 Negative x plus three. 00:25:54.340 --> 00:25:56.780 Let me just leave this x minus three. 00:26:00.240 --> 00:26:01.800 If x is less than three. 00:26:03.360 --> 00:26:05.040 Okay, all absolute value functions 00:26:05.040 --> 00:26:06.520 are piecewise functions, right? 00:26:06.740 --> 00:26:07.800 They all, they look like a V. 00:26:08.580 --> 00:26:09.900 There's two different pieces. 00:26:15.000 --> 00:26:21.080 Okay, so going back here, we're going to have to write a piecewise definition. 00:26:27.620 --> 00:26:29.940 So we're approaching three here, right? 00:26:30.060 --> 00:26:31.700 Not from left to right, but both sides. 00:26:33.500 --> 00:26:36.480 But this thing depends on each side of the three. 00:26:36.480 --> 00:26:42.311 So we're gonna have to break these up into the left-hand limit and the right-hand 00:26:42.311 --> 00:26:43.061 limit. 00:26:43.160 --> 00:26:54.440 So the limit as x approaches 3. 00:26:54.900 --> 00:26:55.860 Start from the left. 00:27:06.860 --> 00:27:14.193 OK, so since we're going from the left, are we always greater than or equal to 3 or 00:27:14.193 --> 00:27:14.943 less 00:27:14.600 --> 00:27:15.500 than 3? 00:27:16.600 --> 00:27:18.380 Less than 3. 00:27:19.280 --> 00:27:22.360 So I'm going to substitute this for this right here. 00:27:23.500 --> 00:27:29.500 So this is the limit as x approaches 3 from the left 00:27:30.920 --> 00:27:37.720 of negative, open parenthesis, x minus 3, close parenthesis, over 3 minus x. 00:27:44.460 --> 00:27:48.660 Which is the limit as x approaches 3 from the left. 00:27:48.660 --> 00:27:59.124 I distribute that negative and switch it, we get 3 minus x over 3 minus x, which 00:27:59.124 --> 00:27:59.874 simplifies 00:27:59.740 --> 00:28:01.880 to 1. 00:28:04.560 --> 00:28:11.980 And now let's take the limit as x approaches 3 but from the right-hand side. 00:28:15.040 --> 00:28:21.120 Now, since we're approaching three from the right, this absolute value. 00:28:26.460 --> 00:28:31.220 Okay, we're to the right of three, so I can substitute this in. 00:28:33.860 --> 00:28:38.720 So this is x minus three over three minus x. 00:28:41.280 --> 00:28:42.320 And... 00:28:46.940 --> 00:28:49.020 How do I simplify this? 00:28:53.100 --> 00:28:54.060 You can factor out the negative. 00:28:57.920 --> 00:29:09.780 Either from the numerator or denominator, we get negative x plus three, which if you 00:29:09.780 --> 00:29:19.640 around they simplify to 1 and now we get negative so the limit from the left 00:29:19.640 --> 00:29:26.480 exists it's equal to 1 the limit from the right exists negative 1 so what is 00:29:26.480 --> 00:29:35.840 the limit as X approaches 3 this does not exist 00:29:46.220 --> 00:29:51.886 Just to get an idea of this function right here, I think it looks something like 00:29:51.886 --> 00:29:52.636 this. 00:29:54.300 --> 00:29:57.280 everywhere to the right. 00:30:00.760 --> 00:30:02.440 It's got a cool function. 00:30:04.500 --> 00:30:09.161 We're to the right of 3, it's always going to be negative 1, or left to be positive 00:30:09.161 --> 00:30:09.911 1. 00:30:10.640 --> 00:30:11.740 Okay, so what's this trick? 00:30:12.640 --> 00:30:16.020 Is we have to break up piecewise functions into two pieces. 00:30:18.200 --> 00:30:20.540 And especially involving absolute value. 00:30:20.540 --> 00:30:24.320 Okay, so break up. 00:30:27.060 --> 00:30:44.387 Okay, especially involving anything involving absolute values is a piecewise 00:30:44.387 --> 00:30:46.120 function. 00:30:48.740 --> 00:30:49.960 All right, what are we going to do here? 00:30:53.720 --> 00:30:58.660 Actually, let's do a, no, let's just do this one. 00:31:02.300 --> 00:31:05.180 Here we cannot just use direct substitution. 00:31:07.220 --> 00:31:13.660 If I use direct substitution, you plug in 1 to the function. 00:31:13.820 --> 00:31:14.860 What is f of 1? 00:31:18.640 --> 00:31:21.720 Do I plug it into here or do I plug it into here? 00:31:23.100 --> 00:31:26.160 The top one, because that's in this domain right here. 00:31:27.100 --> 00:31:32.080 So that's ln of 1, which is equal to 0. 00:31:34.220 --> 00:31:40.399 We can't use direct substitution because, well we can only if we know it's 00:31:40.399 --> 00:31:41.149 continuous. 00:31:41.780 --> 00:31:45.457 And we don't know if this function is continuous because this has two different 00:31:45.457 --> 00:31:46.207 pieces 00:31:46.480 --> 00:31:52.580 Okay, so especially any time it changes right here at whatever number this is 00:31:52.580 --> 00:31:56.755 We got to just break up and do the left-hand limit and the right-hand limit. So 00:31:56.755 --> 00:31:58.060 let's start with the left 00:31:59.120 --> 00:32:00.080 X approaches 00:32:01.240 --> 00:32:03.080 0 sorry 1 00:32:04.480 --> 00:32:05.720 From the left 00:32:18.980 --> 00:32:24.260 Now, since we're always to the left, what is f of x? 00:32:27.100 --> 00:32:28.160 Is x squared. 00:32:28.440 --> 00:32:33.620 So I can write the limit as x approaches 1 from the left of x squared. 00:32:36.220 --> 00:32:38.480 Now this is a continuous function. 00:32:38.480 --> 00:32:41.160 So we can just use direct substitution. 00:32:42.280 --> 00:32:44.860 If we plug it in, we get 1 squared, which is 1. 00:32:50.960 --> 00:32:56.660 As we approach from the 00:32:56.660 --> 00:33:01.260 right of f of x 00:33:10.280 --> 00:33:17.000 Okay, since we're always to the right of one, our function is equal to ln of x. 00:33:18.520 --> 00:33:22.040 So I can substitute in for f of x ln of x. 00:33:25.200 --> 00:33:28.880 And again, this is a continuous function, so we can just plug it in. 00:33:33.180 --> 00:33:40.340 And we get ln of 1 and ln of 1 is 0. 00:33:44.300 --> 00:33:47.480 So the left-hand limit exists, the right-hand limit exists. 00:33:49.720 --> 00:33:52.660 So what is this limit as x approaches 1? 00:33:53.800 --> 00:34:01.040 It does not exist because the left-hand limit does not equal the right-hand limit. 00:34:07.580 --> 00:34:17.620 A more interesting question is: what if the lower piece were x squared plus c? 00:34:17.620 --> 00:34:29.060 The question is, what value of C makes this continuous everywhere? 00:34:29.060 --> 00:34:29.960 everywhere. 00:34:32.340 --> 00:34:33.860 So you look at each piece. 00:34:34.200 --> 00:34:35.400 Is this continuous everywhere? 00:34:38.320 --> 00:34:39.220 Not really. 00:34:40.140 --> 00:34:41.820 It is continuous on what domain? 00:34:42.260 --> 00:34:43.440 Why won't it change colors? 00:34:44.820 --> 00:34:47.440 As long as x is greater than zero, it's continuous. 00:34:48.160 --> 00:34:50.860 And it's only this when it's continuous, or sorry, 00:34:51.220 --> 00:34:52.120 when it's that function. 00:34:52.320 --> 00:34:54.300 So this is continuous everywhere here. 00:34:55.660 --> 00:34:58.200 This is, no matter what c is, is a polynomial. 00:34:59.120 --> 00:35:00.860 So it's definitely continuous everywhere here. 00:35:01.060 --> 00:35:05.940 So the only question is, at 1, we got to make it continuous. 00:35:07.560 --> 00:35:16.040 So we got to, this is. 00:35:21.520 --> 00:35:22.900 This is natural log. 00:35:24.220 --> 00:35:29.160 And then this is just x squared. 00:35:33.220 --> 00:35:37.280 So we just have to shift it up or down so at one they're exactly the same value. 00:35:40.560 --> 00:35:44.860 Long story short, we just need to have the two limits equal. 00:35:47.360 --> 00:35:55.260 With plus c, the left-hand limit would be 1 plus c. 00:35:57.060 --> 00:35:58.700 So to make it continuous, we just 00:35:58.700 --> 00:36:00.780 have to make sure these match. 00:36:02.100 --> 00:36:04.040 So 1 plus C equals 0. 00:36:04.720 --> 00:36:05.840 C is negative 1. 00:36:09.720 --> 00:36:11.940 OK, that was, we're going to get to more 00:36:11.940 --> 00:36:13.180 about continuous functions later, 00:36:13.180 --> 00:36:18.620 but just a little quick question. Let's see.