WEBVTT 00:00:01.001 --> 00:00:06.302 The first topic in the study of calculus is always that of limits. 00:00:06.304 --> 00:00:13.500 We're going to study limits in two parts, first limits from a graph and then limits from an equation. 00:00:13.502 --> 00:00:15.901 So let's find out what is a limit? 00:00:15.903 --> 00:00:25.202 The function f has the limit L as x approaches c if the value f of x, 00:00:25.204 --> 00:00:27.501 which is the y value of the function, 00:00:27.503 --> 00:00:32.302 can be made close to the single real number L whenever x is close to c 00:00:32.304 --> 00:00:36.101 on either side of c but not necessarily equal to c. 00:00:36.101 --> 00:00:41.900 Now that's a lot of words. It's best to do it using notation, and so here's the notation. 00:00:41.902 --> 00:00:46.603 The limit as x approaches c of f of x is L. 00:00:46.605 --> 00:00:51.600 Now there are some notes or comments that need to be made here. 00:00:51.602 --> 00:00:55.902 First of all, the existence of a limit at c has nothing, 00:00:55.904 --> 00:01:00.399 nothing, to do with the value of the function at c. 00:01:00.401 --> 00:01:03.502 That's a little hard to get used to, but you will get used to it. 00:01:03.504 --> 00:01:07.802 In fact, c may not even be in the domain of the function. 00:01:07.804 --> 00:01:12.201 However, the function must be defined on both sides of c. 00:01:12.203 --> 00:01:19.801 Furthermore, the function f has the limit L as x approaches c if 00:01:19.803 --> 00:01:25.400 and only if the limit from the left side of c equals the limit from the right side of c, 00:01:25.402 --> 00:01:27.499 and again we go to notation for this. 00:01:27.501 --> 00:01:32.203 The limit as x approaches c of f of x is L 00:01:32.205 --> 00:01:37.002 if the limit from the left, as x approaches c from the left, 00:01:37.004 --> 00:01:39.901 and that's how the negative sign is read from the left, 00:01:39.903 --> 00:01:44.202 of f of x is L, and the limit as x approaches c from the right, 00:01:44.202 --> 00:01:47.802 and that's what the plus sign is. That means from the right. 00:01:47.804 --> 00:01:50.800 These two limits have to be the same. 00:01:50.802 --> 00:01:56.200 This becomes a working definition, and this is actually how we are going to read limits. 00:01:56.202 --> 00:01:58.901 We're going to look at the limit from the left and the limit from the right. 00:01:58.903 --> 00:02:03.901 One further note: If the limit from the left does not equal the limit from the right, 00:02:03.903 --> 00:02:10.500 then the limit, sometimes called the straight limit, does not exist at all, and so we write DNE. 00:02:10.502 --> 00:02:13.500 Alright, let's take a look at some examples. 00:02:13.502 --> 00:02:20.201 Example 1. Let's take a look at this graph. We want to find the limit as x approaches 2. 00:02:20.203 --> 00:02:25.100 x equals 2 all up and down this vertical line, not just at the tick mark. 00:02:25.102 --> 00:02:37.001 So what we do is we scan across from the left, and what we're looking for here is the limit as x approaches 2 00:02:37.003 --> 00:02:47.701 from the left of f of x. Ok, so we scan from the left and as we get closer and closer to the line x equals 2, 00:02:47.703 --> 00:02:50.799 the value of the function gets closer and closer to 3. 00:02:50.801 --> 00:02:56.600 We're just going to use the whole number 3 for convenience at this point. 00:02:56.602 --> 00:03:07.302 b. Let's find the limit as x approaches 2 from the right of f of x. We do the same procedure where 00:03:07.304 --> 00:03:14.600 we come in and we scan from the right. As we scan in, the values are coming down, down, down, down 00:03:14.602 --> 00:03:23.400 and getting close to 3. So what we see here is the limit from the left is 3, the limit from the right is 3, 00:03:23.402 --> 00:03:34.000 and so, what's sometimes called the straight limit, the limit as x approaches 2 of f of x equals 3 00:03:34.002 --> 00:03:37.200 because the limit from the left equals the limit from the right. 00:03:37.202 --> 00:03:44.898 Let's take a look at example 2. The graph is similar to example 1, but the difference is the function has the 00:03:44.900 --> 00:03:53.999 value at x equals 2 of 1. Let's see how this affects the limit. We read the limit as x approaches 2 from the left 00:03:54.001 --> 00:04:04.200 the same way. We scan in. We see the function is approaching 3, so the limit from the left is 3. 00:04:04.202 --> 00:04:12.795 We do the same thing from the right. We come in from the right, and we see the limit from the right is 3. 00:04:12.797 --> 00:04:17.800 Notice that this value has no effect on the limit whatsoever. 00:04:17.802 --> 00:04:23.900 The limit from the left equals the limit from the right, and so the limit as x approaches 2 is 3. 00:04:23.902 --> 00:04:35.200 What this means here, and this is going to come up later in another topic, is that f of 2, in this example, equals 1. 00:04:35.202 --> 00:04:43.001 Once again, that has nothing to do with the limit. The limit is 3. Let's take a look at another example. 00:04:43.003 --> 00:04:51.102 Again, we have a similar graph, but this time instead of having a hole at (2, 3) or having a point at (2, 1) 00:04:51.104 --> 00:04:55.905 we actually have this point filled in. Let's see how this affects the limit. 00:04:55.907 --> 00:05:03.700 The limit as x approaches 2 from the left, as we come in, is still 3. 00:05:03.702 --> 00:05:10.304 The limit as x approaches 2 from the right, as we come in from the right, is also still 3, 00:05:10.306 --> 00:05:18.402 and so the limit as x approaches 2 is 3. Once again, the actual value of the function at c has 00:05:18.404 --> 00:05:30.400 nothing to do with the limit. We could write f of 2 equals 3, but again that has no effect on the limit. 00:05:30.402 --> 00:05:34.401 Let's take a look at this graph. This one is different from the first three we looked at. 00:05:34.403 --> 00:05:41.299 Let's find the limit as x approaches 2 from the left. As you come in from the left you can see the 00:05:41.301 --> 00:05:49.601 value of the function is rising, rising, rising, rising until it gets close to 1. 00:05:49.603 --> 00:05:57.801 But when you read the limit from the right, you can see that as you get closer and closer with the x-values 00:05:57.803 --> 00:06:04.400 the y-values are going down, down, down and closer to 3. So the limit from the right is 3. 00:06:04.402 --> 00:06:09.399 So what do you have? You have the situation where the limit from the left and the limit from the right exist 00:06:09.401 --> 00:06:19.788 but do not equal a single real number anything, and so therefore this limit does not exist. 00:06:19.790 --> 00:06:27.701 Example 5. This graph is similar to the graph in example 4, but the function is defined at 2 00:06:27.703 --> 00:06:33.000 and has the value negative 1. Let's see what this has to do with our limit. 00:06:33.002 --> 00:06:47.801 The limit as x approaches 2 from the left is 1. The limit as x approaches 2 from the right again is 3. 00:06:47.801 --> 00:06:52.198 Now that means the limit from the left is not equal to the limit from the right. 00:06:52.200 --> 00:06:59.897 Therefore, the limit does not exist. Note that the function has a value here of negative 1. 00:06:59.899 --> 00:07:10.401 So f of 2 is negative 1, and this has no effect on the limit whatsoever. 00:07:10.403 --> 00:07:17.800 The left and the right are not the same. Therefore, this does not exist. Ok, another example. 00:07:17.802 --> 00:07:22.198 Again the graph is similar to examples 4 and 5, but instead of the function having the 00:07:22.200 --> 00:07:27.901 value (2, negative 1) it now has the value (2, positive 1), and it fills in this little hole right here. 00:07:27.903 --> 00:07:32.601 Let's find the limit again as x approaches 2. The limit as x approaches 2 from the left, 00:07:32.603 --> 00:07:40.000 we cruise in again from the left, and we get 1. No change from the previous examples. 00:07:40.002 --> 00:07:46.801 We come in and we test the limit as x approaches 2 from the right, and again we get 3. 00:07:46.803 --> 00:07:52.701 The limit from the left does not equal the limit from the right. Therefore, the limit does not exist. 00:07:52.703 --> 00:08:00.701 Again, this has nothing to do with the fact that the function has at 2 the value of 1. 00:08:00.703 --> 00:08:06.800 We still read the limit only looking at the limit from the left and the limit from the right. 00:08:06.802 --> 00:08:12.501 Ok, example 7. Let's take a look at this graph. The first thing you notice is that you have a 00:08:12.503 --> 00:08:18.401 vertical asymptote at x equals 2 which means the function is not defined at x equals 2. 00:08:18.403 --> 00:08:24.199 You can also see there's a horizontal asymptote at y equals 0. Let's see how this affects the limit. 00:08:24.201 --> 00:08:32.301 Now you have to be careful here because the definition of limit says clearly a single real number L, 00:08:32.303 --> 00:08:38.501 but sometimes there are opportunities where you can use something else here. 00:08:38.503 --> 00:08:44.301 So let me show what we're going to do. We're going to find the limit as x approaches 2 from the left. 00:08:44.303 --> 00:08:48.901 As you come in from the left, you can see that the function is going down, down, down, down, 00:08:48.903 --> 00:08:52.502 down, which means it's approaching negative infinity. Now we all know that 00:08:52.504 --> 00:08:56.901 negative infinity is not a single real number, but the time is going to come when we're 00:08:56.903 --> 00:09:02.802 graphing functions where you need a description or you need to describe the behavior more than 00:09:02.804 --> 00:09:09.700 you actually need the single real number. So the technical answer here is DNE, but the working answer 00:09:09.702 --> 00:09:17.998 here for us is going to be negative infinity because this is going to suit our needs and our purposes. 00:09:18.000 --> 00:09:23.100 Ok, we take the limit as x approaches 2 from the right, and doing the same thing we did before, 00:09:23.102 --> 00:09:29.501 we describe the behavior. Actually there is no single real number, but the behavior here is that the 00:09:29.503 --> 00:09:36.600 function approaches positive infinity, and so that's what we're going to actually write here in the blank. 00:09:36.602 --> 00:09:42.497 You can see that the limit from the left and the limit from the right actually do not exist. Not only 00:09:42.499 --> 00:09:52.201 are they not the same symbol, but they do not exist, and so the limit as x approaches 2 does not exist. 00:09:52.203 --> 00:10:04.000 Also you can observe that f of 2 does not exist, but again that has nothing to do with the limit. 00:10:04.002 --> 00:10:11.300 Ok, one final example. Let's take a look at this graph. We have a vertical asymptote at x equals 2. 00:10:11.302 --> 00:10:19.000 We have a horizontal asymptote at y equals 0. As we read the limit as x approaches 2 from the left, 00:10:19.002 --> 00:10:25.998 we can see that as we come in closer and closer to x equals 2 the function is rising and rising 00:10:26.000 --> 00:10:33.699 and approaching positive infinity. Even though the limit does not exist, we like to describe the behavior 00:10:33.701 --> 00:10:39.701 by using positive infinity. But again, keep in mind the limit does not exist from the left. 00:10:39.703 --> 00:10:47.401 Ok, the limit from the right. As you come in from the right, the behavior is similar to the left. 00:10:47.403 --> 00:10:53.301 The function is rising and this is approaching positive infinity. Again, the limit from the right 00:10:53.303 --> 00:10:59.301 does not exist, but we're going to describe the behavior using positive infinity. 00:10:59.303 --> 00:11:06.101 Ok now, we have an interesting situation here. These are the same symbols, and so you might 00:11:06.103 --> 00:11:14.799 think that because these are the same symbols that means the limit as x approaches 2 is infinity, but 00:11:14.8011 --> 00:11:20.300 remember this limit does not exist, and this limit does not exist, and so therefore 00:11:20.302 --> 00:11:27.301 the limit as x approaches 2 does not exist. 00:11:27.303 --> 00:11:33.000 Ok, so what have we done? We have gotten comfortable with the definition of limits. 00:11:33.002 --> 00:11:38.701 We've learned how to read limits from a graph by looking at the left and the right-hand limits, 00:11:38.703 --> 00:11:41.900 and most importantly, we have learned that 00:11:41.902 --> 00:11:49.301 the actual value of the function at the point has nothing to do with the limit.