WEBVTT
1
00:00:00.000 --> 00:00:08.000
limits and graphs.
2
00:00:08.000 --> 00:00:12.000
Suppose the function f has the graph shown on the right.
3
00:00:12.000 --> 00:00:18.000
Our goal is to describe the behavior of f in the vicinity of x equal to 1 in a concise manner.
4
00:00:18.000 --> 00:00:23.000
Let's first notice that the value f of 1 is equal to 1.
5
00:00:23.000 --> 00:00:29.000
Yet, if x is close but not equal to 1, then f of x is close to 2.
6
00:00:29.000 --> 00:00:34.000
In fact, the closer x is to 1, the closer f of x is to 2.
7
00:00:34.000 --> 00:00:40.000
So the number 2 is crucial in describing the behavior of f near 1.
8
00:00:40.000 --> 00:00:48.000
The way that we describe this behavior is to say that 2 is the limit of f of x as x approaches 1.
9
00:00:48.000 --> 00:00:52.000
This is written compactly in the manner shown.
10
00:00:52.000 --> 00:01:00.000
To be a little more precise, the reason 2 is the limit as x approaches 1 is that, or any interval centered at 2 on the y axis.
11
00:01:00.000 --> 00:01:10.000
No matter how small, the number f of x will be in that interval for all x other than 1 in some sufficiently small interval centered at 1 on the x axis.
12
00:01:10.000 --> 00:01:18.000
Also, we point out that the limit as x approaches 1 has nothing to do with the value of f at 1.
13
00:01:18.000 --> 00:01:25.000
We could change f of 1 to any number we like, or even leave it undefined, and the limit remains 2.
14
00:01:25.000 --> 00:01:33.000
Note that if the limit as x approaches 1 is different from f of 1, there's a hole in the graph at the point 1,2.
15
00:01:33.000 --> 00:01:37.000
If f of 1 were equal to the limit, the hole would be filled.
16
00:01:37.000 --> 00:01:49.000
In fact, at any point where the graph of f is continuous, the y coordinate, that is the value of f, will equal the limit of f as x approaches the x coordinate of that point.
17
00:01:49.000 --> 00:01:55.000
So value and limit coincide wherever the graph of f is continuous.
18
00:01:55.000 --> 00:02:03.000
This idea is the basis of the mathematical definition of continuity that you will see later.
19
00:02:03.000 --> 00:02:09.000
Let's look at another example. Again, suppose that f is the function whose graph is shown on the right.
20
00:02:09.000 --> 00:02:14.000
Here the interesting behavior of the function is in the vicinity of x equals 0.
21
00:02:14.000 --> 00:02:18.000
Let's first notice that the value f of 0 is equal to 2.
22
00:02:18.000 --> 00:02:23.000
If x is close to and less than 0, then f of x is close to 2.
23
00:02:23.000 --> 00:02:30.000
In fact, the closer x is to 0, while x is less than 0, the closer f of x is to 2.
24
00:02:30.000 --> 00:02:35.000
But if x is close to and greater than 0, then f of x is close to 1.
25
00:02:35.000 --> 00:02:41.000
In fact, the closer x is to 0, while being greater than 0, the closer f of x is to 1.
26
00:02:41.000 --> 00:02:47.000
Therefore, there is no number that can serve as the limit of f of x as x approaches 0.
27
00:02:47.000 --> 00:02:50.000
That is, the limit does not exist.
28
00:02:50.000 --> 00:02:56.000
However, we can describe the behavior of f near x equals 0 in terms of one-sided limits.
29
00:02:56.000 --> 00:03:03.000
Eard 2 is the limit of f of x as x approaches 0 from the left or from below.
30
00:03:03.000 --> 00:03:15.000
This means that for any interval centered at 2 on the y axis, f of x will be in that interval whenever x is in a sufficiently small interval whose right endpoint is 0.
31
00:03:15.000 --> 00:03:21.000
1 is the limit of f of x as x approaches 0 from the right or from above.
32
00:03:21.000 --> 00:03:35.000
This means that for any interval centered at 1 on the y axis, f of x will be in that interval whenever x is in a sufficiently small open interval whose left endpoint is 0.
33
00:03:35.000 --> 00:03:39.000
This example illustrates a very important fact about limits.
34
00:03:39.000 --> 00:03:57.000
The limit of f of x as x approaches some number a exists if and only if both of the one-sided limits as x approaches a exist and coincide, that is, if and only if f of x approaches the same number as x approaches a from both the left and the right.
35
00:03:57.000 --> 00:04:04.000
When this happens, the limit equals the common value of the one-sided limits.
36
00:04:04.000 --> 00:04:06.000
Another example.
37
00:04:06.000 --> 00:04:14.000
Again, suppose that f is the function whose graph is shown on the right. Here the interesting behavior of f is in the vicinity of x equal to 2.
38
00:04:14.000 --> 00:04:20.000
Notice that f of 2 is undefined and the line x equals 2 is a vertical asymptote.
39
00:04:20.000 --> 00:04:25.000
If x is close to 2 while less than 2, the net of x is large and positive.
40
00:04:25.000 --> 00:04:31.000
In fact, the closer x is to 2 while less than 2, the larger f of x is.
41
00:04:31.000 --> 00:04:37.000
If x is close to 2 while greater than 2, the net of x is large and negative.
42
00:04:37.000 --> 00:04:45.000
Therefore, there is no number that can serve as the limit of f of x as x approaches 2, that is, the limit does not exist.
43
00:04:45.000 --> 00:04:50.000
In fact, neither of the one-sided limits as x approaches 2 exists.
44
00:04:50.000 --> 00:04:58.000
However, we can describe the behavior of the function near the vertical asymptote in terms of infinite one-sided limits.
45
00:04:58.000 --> 00:05:04.000
Here, we say that the limit of f of x as x approaches 2 from the left is plus infinity.
46
00:05:04.000 --> 00:05:10.000
And we say that the limit as x approaches 2 from the right is negative infinity.
47
00:05:10.000 --> 00:05:26.000
The limit from the left is positive infinity because given any positive number, no matter how large, f of x will be greater than that number, for all x and a sufficiently small open interval whose right end point is 2.
48
00:05:26.000 --> 00:05:39.000
The limit from the right is negative infinity because given any negative number, no matter how large, f of x will be less than that number for all x and a sufficiently small open interval whose left end point is 2.
49
00:05:39.000 --> 00:05:53.000
We should point out that in this example, since the one-sided limits do not agree, the limit as x approaches 2 does not exist, even in an infinite sense.
50
00:05:53.000 --> 00:05:57.000
Now let's look at a few example exercises. This is the first one.
51
00:05:57.000 --> 00:06:16.000
Given the function graphed here, we want to determine a, the value f of 1, b, the limit of f of x as x approaches 1 from the left, c, the limit of f of x as x approaches 1 from the right, and d, the two-sided limit of f of x as x approaches 1.
52
00:06:16.000 --> 00:06:25.000
Because the point 1, 1 is on the graph, we conclude that f of 1 is equal to 1.
53
00:06:25.000 --> 00:06:36.000
As x approaches 2 from the left, the corresponding y coordinates or function values are approaching 2. So the left-sided limit is 2.
54
00:06:36.000 --> 00:06:46.000
As x approaches 1 from the right, the corresponding function values are approaching 0. So the right-sided limit as x approaches 1 equals 0.
55
00:06:46.000 --> 00:07:00.000
Now, since the two-one-sided limits do not agree, the limit as x approaches 1 of f of x does not exist.
56
00:07:00.000 --> 00:07:20.000
This example is similar to the previous one. Given the function graphed here, we'd like to determine a, the value of the function at 2, b, the limit as x approaches 2 from the left, c, the limit as x approaches 2 from the right, and d, the limit as x approaches 2.
57
00:07:20.000 --> 00:07:38.000
f of 2 is equal to 0, because the point 2 is 0 is on the graph. As x approaches 2 from the left, the corresponding y coordinates or function values are approaching 0. So the limit as x approaches 2 from the left equals 0.
58
00:07:38.000 --> 00:07:50.000
As x approaches 2 from the right, function values are negative and getting larger and larger. So the limit as x approaches 2 from the right is negative infinity.
59
00:07:50.000 --> 00:08:08.000
Now, because the two-one-sided limits do not agree, or simply because the right-sided limit does not exist, the limit as x approaches 2 of f of x does not exist.
60
00:08:08.000 --> 00:08:24.000
In this example exercise, we want to sketch the graph of a function f, defined for x greater than minus 1, and less than 3, so the list of conditions concerning limits and or values at x equal to minus 1, 0, 1, 2, and 3 are true.
61
00:08:24.000 --> 00:08:40.000
Let's begin at the left endpoint and work our way from left to right. f of negative 1 is undefined, and the right-sided limit as x approaches negative 1 is 2, so near negative 1, the graph might look something like this.
62
00:08:40.000 --> 00:08:53.000
Since f of 0 is 1, we'll plot the point 0, 1, and since the limit as x approaches 0 is 0, the graph might look something like this between negative 1 and 0.
63
00:08:53.000 --> 00:09:04.000
Next, we plot a point at 1, 0, and we note that the infinite left-sided limit at 1 means that the line x equals 1 is a vertical asymptote.
64
00:09:04.000 --> 00:09:18.000
Since f of x approaches positive infinity as x approaches 1 from the left, and again using the fact that the limit at 0 is 0, we conclude that the graph might look something like this between 0 and 1.
65
00:09:18.000 --> 00:09:27.000
Next, let's use the fact that the limit as x approaches 1 from the right as 1 to continue the graph past 1.
66
00:09:27.000 --> 00:09:41.000
The value at 2 and the left-sided limit as x approaches 2 are both 2, and so we plot the point 2, 2, and then sketch the graph for x between 1 and 2, something like this.
67
00:09:41.000 --> 00:10:00.000
Now, since the right-sided limit as x approaches 2 is 0, we continue the graph past 2 like this, and finally, we finished the graph making use of the fact that the limit as x approaches 3 from the left is equal to 1.
68
00:10:00.000 --> 00:10:14.000
There are many other ways that this graph might have been drawn. Other possibilities need only show the correct values and limiting behavior that x equals negative 1, 0, 1, 2, and 3.
69
00:10:14.000 --> 00:10:32.000
In particular, this is another possibility where the graph wiggles around a bit more between those values of x.