AP Calculus AB — Limits at Infinity and End Behavior
Corrected lesson transcript


## 0:00 — Limits at Infinity and End Behavior

[0:05] Okay, today we're talking about...
[0:07] What are we talking about?
[0:11] We're going to have a quiz on Wednesday, not today.
[0:15] I want you to do this last thing.
[0:19] I want to put a limit as infinity.
[0:37] That means either the limit as x approaches infinity or some function, the limit as x
[0:49] approaches negative infinity.
[0:52] Some other way.
[1:01] This is the exact same thing we did in math 2.3,
[1:05] This is the same idea you studied as the end behavior of a function.
[1:07] It's the exact same thing as the end behavior of a function.
[1:39] Pretend that's some quadratic.
[1:48] The limit as x approaches infinity of x squared is infinity.
[2:02] So the limit as x goes to infinity, we want to know as x gets larger, what is f of x doing?
[2:09] And for quadratics, what's it doing?
[2:13] Getting larger and larger and larger.
[2:15] All polynomials either go up to positive infinity or go down to negative infinity.
[2:20] So our answer here would be infinity.
[2:33] Let's say some fourth degree polynomial.
[2:38] as x goes to infinity of whatever this function is.
[2:48] Again, if it's a polynomial, the answer is either going to be positive or negative infinity.
[2:53] So here, as x gets larger, this keeps going down and down and down and down.
[2:59] So this would be negative infinity.
[3:01] Okay.
[3:20] Oops.
[3:21] Let me do this one.
[3:38] Rational functions often had horizontal asymptotes.
[3:46] Do we remember this?
[3:50] And here, the limit, let's call this g of x, as x goes to infinity, this gets closer
[4:04] and closer to the number 3.
[4:05] It will never be 3, but this has a horizontal asymptote, the answer is 3.
[4:17] And then there's cases like sine of x.
[4:28] Let's say infinity or negative infinity.
[4:32] This is bounded, but will ever go approach a single number?
[4:37] No, so like this, something like this, the answer
[4:39] does not exist. So all the answers to limit as X goes either positive or negative infinity
[4:48] will either be positive infinity if it keeps going up forever, negative infinity if it

## 4:50 — A Constant over Infinity

[4:58] examples. What kind of function is 4 over x? Rational.
[5:07] OK. You can kind of do some math with infinity.
[5:14] It's kind of like direct substitution.
[5:17] OK. 4 divided by a very, very large number is?
[5:24] Yeah. Well, yeah, I won't get there.
[5:27] Sorry. Let's say this is some number, very large number.
[5:31] 4 divided by a very, very large number is a very, very small number.
[5:35] okay and the larger you plug in the smaller gets okay so in terms of limits
[5:41] The larger the magnitude of x becomes, the smaller the quotient becomes.
[5:48] zero and the same is X goes to negative infinity any number over negative
[6:02] infinity keeps getting smaller and smaller and smaller even though it's negative we
[6:07] don't care that is also zero. So when you try the direct substitution even when
[6:17] we're plugging in infinity anytime you have any number over infinity the limit
[6:24] It always goes to zero.

## 6:56 — Rational Functions: Equal Degrees

[6:59] Let me change this to minus.
[7:02] Sorry, let me change this one to minus.
[7:09] Here we got another rational function.
[7:15] If you try plugging in infinity, we get 3 times infinity squared minus infinity plus
[7:25] 2, or 4 times infinity squared minus 5.
[7:35] What is infinity squared?
[7:38] 3. Infinity times 3 is infinity. Infinity minus infinity plus some number 2. What are
[7:58] we doing here is a good question. You can still think of negative infinity plus 2 as
[8:04] negative infinity either way what is infinity minus infinity no we don't know
[8:14] that's an indeterminate form very important infinity minus infinity is an
[8:21] indeterminate form infinity plus infinity is not indeterminate what does
[8:26] that equal that's just infinity a large number plus a large number is a large
[8:33] But a large number minus another large number, we can't really do math with infinity.
[8:41] So that's an indeterminate form.
[8:42] So our point is we need to do something else here.
[8:46] And what are we going to do here?
[8:54] Some of you were probably taught last year just erase this stuff.
[9:01] Can we just erase stuff?
[9:04] No.
[9:07] The answer is sure and poor but why?
[9:12] Because Mr. Williams just said erase it.
[9:14] Erase it.
[9:15] I'm going to use the last 3.
[9:24] Oh, that was canceled.
[9:26] The x goes to infinity of 3 fourths. It's just 3 fourths.
[9:31] Okay, we just can't erase stuff.
[9:35] No, we can't.
[9:39] So what can we do?
[9:44] How can I show what's going to be equal to 3 fourths?
[9:52] You might be able to factor, I don't know if that's going to help.
[9:55] I don't even know if these factor.
[9:57] Yeah, I guess it does.
[9:59] That's not going to help.
[10:02] There's a trick.
[10:05] We should add it to our tricks.
[10:06] Let me copy this real quick.
[10:27] We're going to multiply the numerator and denominator by something.
[10:30] I'll do that.
[10:35] What?
[10:41] Which is just the number one,
[10:42] as long as X is not zero and it's definitely not zero.
[10:46] We're going to infinity large numbers.
[10:49] And I'm gonna choose the smaller of the two degrees.
[10:55] This is a second degree in the numerator,
[10:57] a second degree, they're the same degree.
[10:58] I'm going to choose 1 over x to the smaller degree, but they're the same.
[11:03] And then I distribute that.
[11:06] After multiplying by one over x squared, the expression becomes three minus one over x plus two over x squared, divided by four minus five over x squared.
[11:16] That over x plus 2 over x squared.
[11:36] OK.
[11:38] And then using our limit laws, the limit of a quotient is the quotient of limits, and
[11:43] then the limit of a sum is the sum of limits.
[11:46] We're not going to write all that, because it's just writing lots of different limits.
[11:54] But, let's just look at each single piece.
[11:57] As x goes to infinity, what does 1 over x go to?
[12:02] Zero.
[12:03] Two over infinity squared is two over infinity, which goes to zero.
[12:12] Five over infinity squared goes to zero.
[12:18] So the numerator goes to 3 minus 0 plus 0 and 4 minus 0, the answer is 3 fourths.
[12:38] When the degrees are equal, the limit is the ratio of the leading coefficients.
[12:45] taught right we look at the two degrees they're the same it's just the leading
[12:49] coefficient divided by the leading coefficient but this is why it would
[12:58] anything change if I went to negative infinity? No, the negative infinity that
[13:05] still goes to 0 0 0 we still get 3 4ths. Okay so this rational function has a
[13:14] horizontal asymptote I don't know what it looks like in between but definitely
[13:18] a 3 4ths goes here and here or something and it has some I don't know what it looks like in
[13:28] but something like that

## 13:51 — Rational Functions: Larger Numerator Degree

[13:55] So here's another rational function.
[14:08] Right, this is a fifth degree polynomial as x goes to positive infinity, this goes to
[14:17] positive infinity, this is a third degree polynomial with a negative leading coefficient.
[14:24] It goes to negative infinity.
[14:27] And what is infinity divided by negative infinity?
[14:30] Indeterminate form.
[14:31] Don't know.
[14:34] OK.
[14:34] So, we've got to do some more stuff.
[14:38] So what are we going to multiply by?
[14:40] One over x cubed divided by one over x cubed.
[14:52] I'll show you if you don't choose the smaller of the two what can happen but we distribute
[15:37] As X goes to infinity, 5 over infinity goes to zero, 1 over infinity cubed goes to zero,
[15:46] negative seven over infinity cube goes to zero.
[15:54] Okay.
[15:55] What does infinity square go to?
[16:01] Over negative three.
[16:05] What is infinity divided by negative three?
[16:08] Negative infinity.
[16:11] A positive number, right?
[16:13] Infinity stands for a very, very large positive number
[16:17] divided by negative three
[16:21] is a negative some very large number.
[16:24] So it's a negative, negative, negative.
[16:28] Okay, I chose the smaller of the two degrees.
[16:34] If you didn't, you run into problems.
[16:41] Let me show you what happens if
[16:54] Let's say I chose x to the fifth.
[17:00] You would get 1 minus 5 over x cubed plus 1 over x cubed over negative 3 over x squared
[17:15] minus 7 over x cubed.
[17:20] You don't have to write this down.
[17:23] So x goes to infinity, this goes to zero, this goes to zero, this goes to zero, this
[17:29] goes to zero, and we get one minus zero, plus zero over zero, plus zero, one over zero.
[17:47] And what does a number over zero go to?
[17:50] In terms of limits.
[17:54] Either positive or negative infinity.
[17:57] But what we don't know here, since we did it here, we don't know, right, the denominator goes to zero.
[18:05] But we don't know if it's from the left side, which is a negative number, or from the right, it's a positive number.
[18:12] So we don't know what this ends up as positive or negative infinity.
[18:18] OK, so we don't want to do this. OK, so anyways, the point was,
[18:27] do one over the smaller of the two degrees and then it will show you whether it's positive or negative infinity.
[18:32] Let's try another one.

## 18:34 — Rational Functions: Larger Denominator Degree

[18:39] Again, if you plug in infinity, you get infinity over infinity indeterminate form.
[18:43] So what am I going to multiply by?
[18:47] The numerator has degree one and the denominator has degree two, so multiply by one over x.
[18:51] So one over x, one over x.
[18:53] A smaller two degrees.
[19:25] this goes to 0, this goes to 0
[19:30] 1 plus 0
[19:33] this goes to negative infinity
[19:38] minus 0
[19:42] what is 1 over negative infinity go to?
[19:46] any number over positive infinity goes to 0
[19:57] So these are three different cases.
[19:59] One here, when the degree in the denominator is greater, this always ends up going zero,
[20:05] but this is kind of why we showed it.
[20:09] Here when the degree in the numerator is larger, it's either going to go to positive or negative
[20:14] infinity.
[20:15] You have to use this to figure out which one and then when the degrees are the same
[20:23] It is always the leading coefficient divided by the leading coefficient
[20:28] And this was why it's the case
[20:34] Okay, so that was rational functions

## 21:02 — Exponential End Behavior

[21:02] What type of function is this?
[21:06] Exponential.
[21:09] Okay, you can kind of do infinity math of this
[21:14] Three times infinity
[21:18] Three times infinity is still infinity. What is e to a very very large number?
[21:28] Like what's e to a thousand at which is not very large
[21:33] It's an enormous number.
[21:36] Big, big, big, big number.
[21:38] Okay, this is infinity.
[21:47] And then, let's see what happens as we go to negative infinity.
[22:17] What is e to a negative exponent?
[22:31] It is 1 over e to positive infinity.
[22:36] Okay, e to positive infinity is infinity.
[22:42] What is 1 over infinity?
[22:45] Zero.
[22:45] It goes to zero.
[22:47] Okay.
[22:49] All exponential functions are either the 3x,
[22:55] looks something like this.
[22:57] All exponential functions on one side,
[22:59] it goes to either positive or negative infinity.
[23:01] On the other side, we always have a horizontal asymptote
[23:03] of zero.
[23:09] So that's the end behavior for all exponential functions.
[23:13] If this were like negative 3x, it would just be flipped.
[23:17] this will go to 0 and this would go to infinity. So you do got to be careful to sign this up.

## 23:19 — Radical Limits and Conjugates

[23:19] And then we'll do the other case.
[23:23] For the radical expression, first evaluate the negative-infinity case.
[23:34] Nine times negative infinity squared plus negative infinity, well forget that.
[23:43] This is a quadratic.
[23:45] Where does a quadratic go as you go to negative infinity?
[23:52] It goes to positive infinity, right?
[23:54] That quadratic looks something like this.
[23:58] So the inside here goes to positive infinity.
[24:02] The square root of positive infinity is... square root of a very big number is still
[24:10] a really big number.
[24:11] So that goes to infinity.
[24:12] And then we got minus 3 times negative infinity.
[24:19] What does negative 3 times negative infinity go to?
[24:22] Infinity.
[24:23] And so our answer is infinity.
[24:29] This is infinity.
[24:34] Now the other one is much more interesting.
[24:47] Here if you plug in infinity, we get infinity minus infinity, which is what?
[24:56] In determinant, in terms of limits it's an indeterminate form.
[25:02] Ok so what are we going to do here?
[25:08] It's one of our old tricks.
[25:12] We're going to multiply by the conjugate, thank you whoever said that.
[25:17] Okay, so we're going to go the square root of 9x squared plus x plus 3x.
[25:46] plus x. The 2 milli terms cancel out, we get minus 9x squared. And now what happens? These
[26:16] cancel.
[26:24] x over x squared.
[26:33] Which we plug infinity, we still get infinity over infinity.
[26:46] We're either going to multiply by 1 over x, or we're going to factor out an x out of here.
[26:53] Let's do this one.
[27:04] We're running out of time, so I've got to speed this up.
[27:19] After rationalizing, divide by x and move the positive x into the square root as the square root of x squared.
[27:26] Is x equal to the square root of x squared?
[27:34] This is very important.
[27:38] The identity x equals the square root of x squared is valid here because x approaches positive infinity.
[27:46] If x is a negative number this is not true.
[27:50] In fact, if x is a negative number, we have to put a negative here.
[28:02] Oh, it's about to ring.
[28:03] So which one is it?
[28:04] Is it a positive or a negative?
[28:05] Which one am I going to substitute in?
[28:08] Since we're going to positive infinity, I'm going to substitute this x right here for
[28:13] this, we get the limit as x goes to infinity.
[28:22] Get rad nine x squared plus x over rad x squared,
[28:30] which now I can make one big square root.
[28:41] We get the square root of 9 plus 1 over x plus 3.
[28:51] Hold on one sec.
[28:54] Now as x goes to infinity this little guy goes to zero.
[28:58] As x approaches infinity, one over x tends to zero, so the limit is one sixth.
[29:05] Okay, be real careful when you bring in
[29:09] into square roots.
[29:11] Sometimes this, sometimes that.
[29:13] Okay, the homework is some delta math.