AP Calculus AB — Limits from Tables and Graphs
Corrected lesson transcript


## 0:00 — Limits from Tables and Graphs


## 0:04 — What a Limit Measures

[0:04] Okay, so today we're talking about limits.
[0:12] What is a limit?
[0:17] this okay you don't have to write this
[0:23] first sentence down actually you don't really you don't need to write this down
[0:26] yet so I put it in there my calculator some function you don't know what it is
[0:36] secret I put into y1 and here's what I want to know I want to know what is our
[0:42] function approaching as X approaches the number 2 okay I don't know what it looks
[0:55] like but as as we plug in values closer and closer to X what are these values
[1:09] getting close to? What is this number? Okay, and as we approach from the other side, what
[1:17] is this approaching? Okay, so this is the whole idea of a limit is as x gets closer
[1:26] to some number, what is our function value getting closer to? Okay, and this is the notation
[1:36] we use. LIM stands for limit. And this is x, we have a little arrow always approaches
[1:43] some number, we put a number here. And sometimes you'll see a positive or a negative sign,
[1:50] and also so that means, and of this function.
[1:54] So we wanna know what x approaches two,
[1:57] as x approaches two,
[1:58] so what numbers should I plug into my calculator?
[2:09] Kind of like we did on the quiz.
[2:12] We kept plugging in numbers closer and closer to 4.
[2:14] So let's try like 1.9.
[2:21] 1.99.
[2:26] 1.999.
[2:47] So I'm plugging in closer x values closer and closer to 2, right?
[2:53] And what does it look like after x is getting closer and closer and closer to 2?
[2:56] 7.
[2:58] OK.
[2:59] So whatever this function is, we think
[3:10] this is getting closer and closer to 7,
[3:15] even though we do not know what happens at x = 2.
[3:18] For the limit, we do not care what happens exactly at x = 2.
[3:21] All we care is what's happening as we get closer to it.
[3:23] So what we actually just found, where's my cap?

## 3:24 — Left-Hand, Right-Hand, and Two-Sided Limits

[3:24] We actually only approached it from the left side,
[3:27] numbers less than two.
[3:30] Okay, we also need to approach it from the right side,
[3:33] numbers greater than two.
[3:35] So I'm gonna plug in, oops.
[3:35] Okay, so as we plug the numbers from the right, the numbers are slightly greater than two
[3:42] and the closer we got to two, what is our function getting closer and closer to seven
[3:48] as well?
[3:51] Okay, limits, we don't care what happens at this number two, we only care what's happening
[3:57] from the left and from the right.
[4:01] Okay, so whatever this function is, and so we use certain notation.
[4:09] If we approach from the left, we put a little, it looks like to the power of a negative sign.
[4:18] Okay, this is me, we're approaching x is approaching 2, but number is less than 2 of this function.
[4:28] Okay, and the answer was 7.
[4:33] If we approach from the right side, it's called the right-hand limit, this was also 7.
[4:42] Okay, so limits, all we care about is what's happening as x gets closer to some number,
[4:49] both from the right and from the left.
[4:52] If these both are the same number, this limit, which is just approaching 2, not from the
[5:00] left, not from the right, but both sides, is equal to 7 as well.
[5:05] So this is called a left-hand limit.
[5:11] It's called the right-hand limit.
[5:16] limit, okay, and if the limit as x approaches whatever number this is, from the left equals
[5:35] the limit as x approaches a from the right, then the limit as x approaches a, which is
[5:54] from the left and right.
[5:58] Okay, so if these are both some number let's call it L the limit then this is also equal to L
[5:59] And we do not care what f(2) is.
[6:04] And we find limits we don't care what f of that of this value is. All we care
[6:10] about is what's happening from the left and from the right. In fact, what is f of 2
[6:23] here? It does not even exist. That is okay—we do not care about f(2) while finding this limit.
[6:31] Later, we will care because the function value helps tell us whether the function is continuous.
[6:36] For now, one way to find a limit is to plug in values just to the left and right.
[6:42] we're finding limits one way is this way you keep plugging in values if you're
[6:46] we're finding limits one way is this way you keep plugging in values if you're
[6:46] given the equations just the left of it just the right of it and they keep
[6:51] getting closer to some number that is the limit what was this secret function
[6:58] I think I know.
[7:01] No idea.
[7:05] Let's see what it is.

## 7:18 — Holes, Vertical Asymptotes, and Rational Functions

[7:18] But what kind of function was this?
[7:21] What do you call this kind of function?
[7:25] Rational!
[7:27] Thank you!
[7:31] What is a rational function?
[7:36] A rational function is a ratio of two polynomials.
[7:55] For example, x squared over e to the x is not a rational function. It is not just a
[8:05] ratio of functions or expressions. It must be a ratio of two
[8:13] polynomials. Thank you. As long as both are polynomials, it is called a
[8:18] rational function. As a review, rational functions have one of two
[8:24] things they either have a vertical asymptotes or what's the other option
[8:37] Oh, well, where is this?
[8:49] Okay, so when values that make the denominator zero will either be vertical asymptote or
[8:56] a hole in the graph.
[8:58] Thank you.
[8:59] Okay.
[9:01] Does this function have a hole in the graph or a vertical asymptote?
[9:06] A hole.
[9:07] A hole.
[9:09] Okay.
[9:09] At two, it doesn't look like this.
[9:13] It looks like a hole in the graph.
[9:24] How can we tell from here that it has a hole?
[9:30] That's a hole in the graph.
[9:32] If this wasn't here, it would be a vertical asymptote.
[9:37] What would happen, in fact, let me put that in the calculator.
[9:54] OK, so now it no longer has a hole.
[10:01] There's going to be a vertical asymptote at x equals 2.
[10:07] What's going to happen when I plug in numbers close to two now?
[10:17] Any guesses?
[10:37] Okay, what's happening with the y-value?
[10:43] Just keeps getting larger except it's negative, so it's really getting smaller and smaller
[10:47] smaller and smaller and smaller.
[10:50] Okay, in fact, and then what's if I plug in numbers greater than two?
[11:07] I like 2.01.
[11:14] 2.001.
[11:21] 2.001.
[11:25] It just keeps getting larger and larger and larger.
[11:27] OK, so if this happens on the calculator,
[11:30] if you're given a secret function,
[11:35] you're plugging it back.
[11:36] Or you could plug in the number.
[11:37] It doesn't matter.
[11:39] Let's talk about that limit, actually.
[11:42] Ch-ch-ch-ch.
[11:50] So this is my function.
[11:53] Was it x squared plus 3?
[11:56] No—minus 3.
[11:56] So the function is (x squared minus 3) divided by (x minus 2).
[11:58] Okay, and I can either write f of x here or I can write in the whole thing
[12:07] Okay, so we plug in values they kept going
[12:11] Smaller or smaller so we say
[12:16] The left-hand limit is negative infinity.
[12:18] Some people say the limit does not exist here, but for this one-sided limit we write
[12:32] negative infinity. Now, what is the two-sided limit
[12:40] as x approaches 2 of this function. If this does not equal this, it does not exist.

## 12:55 — Read Limits from a Graph

[12:55] So today we're only talking about limits based on graphs of a function or based on tables
[13:00] of values. The last example is table values get plugging in numbers or you're
[13:06] given the tables of the X and the Y values. Friday we'll start learning based
[13:17] on equation or yeah function equations. Alright first question what is the limit
[13:24] or let me write down here.
[13:27] The limit as x approaches 2 from the left g of x.
[13:43] So as x gets closer and closer to 2 from the left, what is this getting closer and closer
[13:49] and closer to?
[13:50] 3.
[13:51] so this limit equals 3 the limit as X approaches 2 from the right as X gets
[14:08] closer and closer to f of x gets closer and closer to 1. And what is the limit as x approaches
[14:26] 2? It does not exist. We don't have to look at anything except this number does not equal
[14:35] this number, the left-hand limit does not equal the right-hand limit, the limit as x
[14:40] approaches 2 does not exist.
[14:48] And what is g of 2?
[14:57] Where is the red dot at 2?
[14:59] There is none.
[15:00] So this does not exist as well.

## 15:03 — Limits, Function Values, and Continuity

[15:03] So at every x value, we always have a left-hand limit,
[15:06] a right-hand limit, the limit,
[15:10] and the actual function value.
[15:13] And they all might be a number, they all might not exist.
[15:16] So let's look at five here.
[15:35] The limit as x approaches five from the left.
[15:48] As x approaches 5 from the left, these values get closer and closer to 2.
[15:58] This is the limit as x approaches 5 from the right.
[16:06] as x approaches 5 from the right or g of x approaches 2.
[16:19] So what is the limit as x approaches 2?
[16:24] 5?
[16:25] 1.
[16:27] This?
[16:28] 2.
[16:28] Sorry, 2.
[16:32] My bad.
[16:34] Five. My bad.
[16:39] Since the left-hand limit
[16:41] equals the right-hand limit, so the two-sided limit is exactly 2.
[16:49] And what is G of five?
[16:53] One.
[16:56] Okay.
[16:57] Is our function discontinuous at 5?
[17:02] Yeah, definitely.
[17:05] OK. And how can we tell?
[17:09] Well, obviously we look at it. But if all I gave you is this information,
[17:14] the way we tell if it's continuous is does this equal the limit?
[17:21] OK, if the function value equals the limit,
[17:25] the function is continuous. If the function value does not equal the limit, it is discontinuous.
[17:31] And then our previous example
[17:33] In our previous example, the function value did not exist, so it was definitely discontinuous.
[17:38] But this doesn't equal well this doesn't exist either, so
[17:43] But in fact that's going to be the definition of
[17:47] A function is continuous at a point when the limit equals the function value.
[17:54] We'll define that later.
[17:55] Okay, let's do another one.
[18:05] What is the limit as x approaches 0 from the left?
[18:18] So as x approaches zero, f of x gets closer and closer to, let's call it 2.5.
[18:32] The limit as x approaches zero from the right, as x approaches zero from the right, f of
[18:44] x gets closer and closer to 2.5.
[18:50] What is the limit as x approaches zero?
[18:54] 2.5. 2.5.
[19:02] And what is g of 0? 2.5.
[19:07] 2.5. Is it continuous at 0?
[19:13] Yeah. Okay.
[19:17] And in fact, if we know the function is continuous,
[19:21] Let's say a 2 and 5 are its only discontinuities.
[19:32] Or let me ask you this.
[19:34] the limit as x approaches zero,
[19:45] x to the fifth minus seven x plus four.
[19:53] Is this a continuous function?
[19:59] What do we call this type of function?
[20:02] Polynomial.
[20:03] A polynomial.
[20:05] Okay.
[20:06] All polynomials are continuous over all real numbers.
[20:10] Okay.
[20:12] Since we know it's continuous over all real numbers, we know this limit exactly equals
[20:19] whatever f of zero is.
[20:22] Let's call this f.
[20:24] And what is f of zero?
[20:26] It will be four.
[20:29] Okay.
[20:29] So if a function is continuous at a value—or over all values—
[20:34] the limit as x approaches that value equals the function value.
[20:40] okay, so
[20:45] Like at four here
[20:50] It's definitely continuous there so the limit as X approaches four from the left
[20:59] is going to equal the limit as x approaches 4 from the right.
[21:07] It's going to equal the limit as x approaches 4.
[21:14] It's going to equal g of 4, whatever that number is.
[21:18] Let's call it 1.8. I'm just making up a number.
[21:22] If the function is continuous, the left-hand limit equals the right-hand limit,
[21:26] the limit equals the function value.
[21:31] So every other number here, besides two and five,
[21:34] the limit from the left will equal the limit from the right.
[21:37] The limit will equal the function value.
[21:41] Make sense so far?
[21:42] Okay, if you're getting a graph, it's real easy.

## 21:44 — A Hole Hidden by Graphing Technology

[21:44] Okay, in fact, actually let's do this example.
[21:51] What is the limit as x approaches, let's go to two.
[21:55] Oh no, let's go one.
[21:56] The limit as x approaches 1 from the left is 1.
[22:03] It keeps getting closer and closer to 1.
[22:05] As x approaches 1 from the right, 1.
[22:09] What is the limit as x approaches 1?
[22:12] 1.
[22:13] What is f of 1?
[22:15] 1.
[22:16] No.
[22:18] Ok.
[22:19] We have a hole in the graph right here.
[22:23] You can never tell on Desmos or the graphing calculator there's a hole so be real careful
[22:30] Okay, because it's so small no matter how far I zoom in here
[22:36] Where is it?
[22:38] You will never see the hole
[22:41] Okay, so be real careful on
[22:44] graphing calculators on Desmos
[22:46] You will never see holes in the graph
[22:50] Okay, that's why when we draw them
[22:53] Like here we make the whole giant so, you know, it's a whole that makes sense. Okay

## 23:03 — Jump Discontinuities and One-Sided Continuity

[23:03] You don't have to draw this
[23:06] We'll just do it real quick.
[23:13] What is the limit as this is 0, 0, right?
[23:20] As x approaches 0 from the left, let's call this f.
[23:28] As x approaches 0 from the left, f of x approaches?
[23:33] Maybe 1.
[23:33] What is the limit as x approaches 0 from the right?
[23:42] As x approaches 0, this approaches negative 2.
[23:47] So both of these limits exist.
[23:50] Anytime you get a number it exists.
[23:53] And what is the limit as x approaches 0?
[23:59] It does not exist.
[24:02] Why?
[24:03] because the left-hand limit does not equal the right-hand limit. Okay and what is F of 0?
[24:12] negative 1. Okay is it discontinuous at X equals 0? Definitely, that's some sort of
[24:21] jump discontinuity we call it. We'll talk later, we actually say it's continuous from the left
[24:30] because the left-hand limit equals the function value.
[24:35] It's discontinuous from the right because the limit from the right does not equal the
[24:42] function value.
[24:43] But overall it's definitely discontinuous at zero.
[24:46] What is the limit as x approaches 2 from the left?
[24:56] 2.
[24:58] The limit as x approaches 2 from the right?
[25:03] 0.
[25:05] The two-sided limit as x approaches 2 does not exist because the left-hand
[25:15] limit does not equal the right-hand limit. And f(2) = 1.
[25:25] It's definitely discontinuous at 2. Is it continuous on the left? No because the
[25:32] left-hand limit does not equal the function value.
[25:36] Is it continuous from the right?
[25:37] No, because the right-hand limit,
[25:40] zero does not equal to the function value.
[25:43] Okay.
[25:44] Unlike at x = 0, it is not continuous from the left.
[26:00] What is the limit as x approaches 4 from the left?
[26:10] This gets closer to 3.
[26:13] 3, limit as x approaches 4 from the right.
[26:17] 3, 3.
[26:19] What is the limit as x approaches 4?
[26:22] 3.
[26:24] And what is f of 4?
[26:26] 3.
[26:27] 3.
[26:29] Okay.
[26:31] Is the function continuous at 4?
[26:33] Yes.
[26:34] Yes.
[26:34] If we weren't given a graph, if this limit equals the function value, it's continuous.
[26:43] Okay, and we're going to define later.
[26:45] So that means it's continuous at a point, but later it's continuous on an interval.
[26:51] Okay, I'm hoping you're getting the point.
[26:54] Let's do, I think I got one more for you.

## 26:57 — Infinite Limits with tan x

[26:57] Anybody know what function this is?
[27:00] This side.
[27:01] This is tangent.
[27:09] What is the limit as x approaches pi from the left?
[27:17] Zero.
[27:19] What is the limit as x approaches pi from the right?
[27:23] Zero.
[27:26] What is the limit as x approaches pi?
[27:29] Zero.
[27:30] Zero.
[27:31] As x approaches pi over 2 from the left, the tangent values grow larger and larger.
[27:37] and larger.
[27:39] So we put positive infinity.
[27:48] Now consider the limit as x approaches pi over 2 from the right.
[27:57] It approaches negative infinity; from the right, the graph keeps going down
[28:02] forever. The two-sided limit as x approaches pi over 2 does not exist
[28:13] because the one-sided limits are not equal. If both approached positive infinity, we would write
[28:20] positive in here, they both go down.
[28:22] But they go in different direction,
[28:24] definitely doesn't exist.
[28:32] So is it continuous at pi over two?
[28:37] Definitely not, in fact, what is tan of pi over two?
[28:44] Does not exist.
[28:47] Okay.
[28:50] What do we call this type of discontinuity?
[28:56] We just call it infinite discontinuity.
[29:01] A hole in the graph we call a removable discontinuity.
[29:07] If it jumps, it's called a jump discontinuity.

## 29:09 — Oscillation: sin(pi/x) Near Zero

[29:09] OK.
[29:11] This is a function of sine of pi over x.
[29:21] OK.
[29:21] It's continuous everywhere except to what?
[29:26] At x = 0 the function is undefined, but for every other x we can evaluate sine of that value.
[29:33] The interesting part the limit has
[29:37] X approaches zero
[29:44] Okay to get closer to closer to zero what's happening
[29:51] So let's look at this as x goes to zero,
[29:55] what's happening to this?
[29:59] What's pi over a very, very small number?
[30:04] Like pi divided by 0.00001 is a very big number.
[30:10] The closer x gets to zero, the larger pi divided by x becomes in magnitude.
[30:15] And what is sine of a very, very big number?
[30:21] It just keeps doing this forever.
[30:23] It keeps oscillating.
[30:26] As x approaches zero, the inside approaches infinity in magnitude, and the
[30:31] inside just keeps oscillating back and forth.
[30:34] You get this really cool path.
[30:35] So this, both from the left and from the right, don't approach a single number.
[30:42] It never exceeds 1 or goes below negative 1, but
[30:46] It does not approach a number, so this does not exist.
[30:51] It's called an oscillating discontinuity.
[30:54] It's actually really cool if you look at the on Desmos.
[31:20] The more I zoom in, the more there are.
[31:27] It completely fills up the whole screen.
[31:30] In reality, there is a tiny gap between every one of those lines.
[31:35] But the closer you get, the faster it goes back and forth, back and forth.
[31:39] Super cool.
[31:42] Okay, very weird as well.
[31:46] But there we go.

## 31:50 — Lesson Summary

[31:50] Okay I think we're going to stop here.
[31:57] Again all the limits, the idea of left-hand limit, right-hand limit, the limits, it's
[32:03] only based on tables of values and graphs and Friday we'll talk about when
[32:08] we are given the actual equation.
[32:11] We're just going to, I put a new Delta Math assignment up, so just do Delta Math.
[32:16] So you can get going on that.