AP Calculus AB — Special Trigonometric and Piecewise Limits
Corrected lesson transcript


## 0:00 — Special Trigonometric and Piecewise Limits


## 0:04 — Combine Fractions, Then Rationalize

[0:05] All right. So what's the first thing we always try? Direct substitution. Over here,
[0:21] we get 1 over (0 times the square root of 1 plus 0), minus 1 over 0.
[0:35] 0 go to? Well, in limits, it will either go to positive or negative infinity. It depends
[0:46] on the left to the right. So let's not worry about the left and right phenomenon. So this
[1:08] one over zero goes to either positive or negative infinity. This goes to either positive or
[1:12] negative infinity okay and we get this what is infinity minus infinity not zero
[1:22] we don't know what it is it's another indeterminate form okay if these both go
[1:36] to positive infinity or sorry let's say the one on the left goes to positive and
[1:40] this goes to negative.
[1:42] This would be infinity plus infinity,
[1:44] which does go to infinity.
[1:47] But we don't know,
[1:48] because we didn't do it from the left and the right.
[1:53] So anytime you get infinity minus infinity,
[1:57] indeterminate, we gotta do some more work.
[2:01] All right, so this next trick,
[2:05] is we just need to combine them into one fraction.
[2:08] So we get a fraction minus another fraction.
[2:11] We combine them into one fraction.
[2:13] So we just need to get a common denominator
[2:17] of the two fractions.
[2:27] So we just need to multiply this fraction by something. What do we multiply it by?
[2:34] So we just need to multiply this guy by something.
[2:37] What do we multiply this guy by?
[2:41] I have 1 plus x over 1 plus x.
[2:46] Now they have a common denominator.
[2:50] And now if we try direct substitution, we get 1 minus the square root of 1 plus 0, over 0.
[3:03] Now we get 0 over 0, which is?
[3:07] A different indeterminate form.
[3:09] we're gonna do some more work so now what so we go back to one of our other
[3:22] tricks multiplied by the conjugate because we got two terms here especially
[3:28] one involving square root. So let's multiply by 1 plus 1 plus x. And again, the bottom
[3:51] or the denominator, which we did not choose the numerator, don't multiply that out. Just
[3:55] keep that all factored. Yeah. Again the two middle terms of the conjugate add to be zero.
[4:10] So we get 1 plus 0 minus 1 plus x.
[4:21] Now we distribute this guy.
[4:26] And now what?
[4:32] one of our other tricks.
[4:42] cancel the common factor x.
[4:48] And now we return to direct substitution and see what happens.
[4:53] The denominator becomes 1 times 1 plus 1, so the limit is negative one-half.
[5:01] Okay, so.
[5:06] So the very first trick was to combine multiple fractions or the sum or
[5:12] difference of fractions into one fraction.
[5:17] And then also the other trick is keep trying more tricks.
[5:30] Another technique.
[5:45] Alright.
[5:46] Alright.

## 5:48 — The Special Limit sin(x)/x

[5:50] Direct substitution: what is sine of zero?
[6:01] Oh what do we do here okay this next
[6:17] trick very important one you just memorize the answer memorize okay this
[6:33] is one of the ones we're just gonna memorize the answer is one here it's not
[6:38] obvious
[6:42] Ok, so here is our function sine of x and x.
[6:47] As you approach zero, these two functions get closer and closer and closer to each other.
[6:54] And when you divide two numbers that are very, very close,
[6:58] it gets closer and closer to 1.
[7:04] If I zoom in here, the closer you get to 0, the closer
[7:09] you can tell the difference between the two.
[7:13] And then if I show you this function, y equals sine x.
[7:31] It's the green function.
[7:34] There is a hole right there at x equals zero.
[7:37] That function doesn't exist at x equals zero.
[7:41] However, you approach from the left and the right, it gets closer and closer to one.
[7:45] So the trick is you just memorize this one.
[7:50] Okay.
[7:53] Okay. One of the limits we memorize is this one: sine of x over x.
[7:59] Sine of x over x.
[8:00] O-Rex...

## 8:03 — The Reciprocal Limit x/sin(x)

[8:04] What about this?
[8:10] Is this also going to equal one or no?
[8:13] It's going to be negative.
[8:16] That's right.
[8:17] It's just a guess. That's not a bad guess.
[8:21] Not quite true, but it's not a bad guess.
[8:25] Is this going to equal one? Why or why not?
[8:31] Is this true?
[8:34] wrote down here yes okay right if you had this you take the numerator it's one
[8:46] multiplied by the reciprocal and you get exactly that okay so this limit is the
[8:54] limit as x approaches zero of one divided by sine of x over x. And one of the limit
[9:08] laws that I handed out is the limit of a quotient is the quotient of limits. So we could write
[9:22] this as the limit as X approaches 0 of 1 divided by the limit as X approaches 0
[9:31] sine of X over X and what is this limit? Limit of a constant is just that constant
[9:48] And this limit we just memorized is 1 is definitely equal to 1.
[9:58] Long story short, limit as x approaches 0 of sine of x over x or x over sine of x is
[10:05] 1.
[10:09] Very important.

## 10:12 — The Cosine Form (1-cos(x))/x

[10:14] So if we try direct substitution,
[10:15] we get 1 minus cosine of 0 over 0.
[10:21] What is cosine of 0?
[10:25] 0 over 0.
[10:37] So, is this also going to equal one?
[10:45] It's going to be one?
[10:48] Some people shake their head no?
[10:52] We are also going to memorize this one, but let's figure it out before we get the answer.
[11:01] Here's a little trick.
[11:04] We're going to multiply by 1 plus cosine of x.
[11:12] which is just a conjugate of the numerator there.
[11:21] These add to be zero.
[11:31] Does that help anything?
[11:34] You direct substitution, you still get 0 over 0.
[11:39] However, I could substitute this for something.
[11:46] What is 1 minus cosine squared always equal to?
[11:49] Sine squared, right?
[11:50] Because sine squared plus cosine squared equals 1.
[11:58] So sine squared is one minus cosine squared.
[12:15] If you try direct substitution, we still get zero over zero.
[12:20] Any ideas what we could do from here?
[12:26] What's that?
[12:37] I'm not sure we could do that.
[12:43] However, sine squared is sine times sine, correct?
[12:49] We have an x down here, so we're going to rewrite this as this.
[12:55] sine of x over x times sine of x over 1 plus cosine of x.
[13:17] And what is the limit of this guy, guys?
[13:22] So the limit of a product is the product of limits.
[13:27] So I'm going to rewrite like this.
[13:32] this
[13:33] limit is equal to 1 and now we directly substitute and we get sine of 0 which is
[13:46] 0 1 plus cosine of 0 which is 1 which is 0 over 1 plus 1 which the whole thing is
[14:03] simply zero. Okay, and what we typically do, or what I typically do, instead of writing
[14:24] the limit of the quotient, a limit times a limit, this is technically what's happening.
[14:27] what we typically do is just we say this thing goes to 1 and the limit x goes to 0 of
[14:39] sine of x over 1 plus cosine of x and then we get 0 over 1 or 2 so if you ever
[14:47] see me like put arrows here that's just meaning whatever this function is it
[14:53] goes to whatever the limit of it goes to that number just kind of saves you time
[14:58] from writing the limit times limits on okay so this is another one we memorize
[15:06] this is really the only other one we memorize
[15:13] This is just slightly different.
[15:16] How is this different than 3 to the 1?
[15:23] The last one was 1 minus cosine of x, this is cosine of x minus 1.
[15:31] Is this also going to be equal to 0?
[15:44] Take a guess yes or no everybody give me a thumb.
[15:53] Okay let's work it out it does equal to one I'm sorry does equal to zero.
[16:01] If I factor out a negative out of that, I get the limit as x goes to zero.
[16:11] And then one of our limit laws if we have a coefficient in front what can I do with that?
[16:18] You could distribute what are my limit laws here?
[16:28] Or wasn't at the very top yeah, here it is
[16:37] This one right here.
[16:40] You have a coefficient in front, what can I do with the coefficient?
[16:44] We can bring it outside the limit right here.
[16:55] So this can come out.
[16:56] we get negative 1 times this limit and then let me rewrite these two
[17:10] negative cosine plus x plus 1 is 1 minus cosine of x
[17:20] and what's this equal to
[17:25] Zero, one of the ones we memorized.
[17:27] We just did, so it's negative one times zero, it is zero.
[17:45] You don't really have to memorize it, but again, the reciprocal of this is the same
[17:50] value and if these are swapped it's the same value. They're both 0, they're both 1
[18:00] Let me just write that down anyways.
[18:08] x over sin x
[18:13] also equals to
[18:14] one minute
[18:19] So the reciprocal has the same value here.
[18:22] And we can swap that.
[18:23] I'm going to do some algebra.
[18:39] All right, next one.

## 18:42 — Rewrite Trig Functions Using Sine and Cosine

[18:43] I'll try direct substitution.
[18:47] We get 0 times secant of 0 times cosecant of 0.
[18:57] What is secant of zero?
[18:59] What is cosine of zero?
[19:07] Which is one.
[19:10] And cosecant of x?
[19:14] What is sine of zero?
[19:17] Zero. So one of this does not exist.
[19:25] So, we got to do something else.
[19:27] What are we going to do?
[19:31] Any ideas?
[19:36] Figuring out secant, cosecant, cotangent, they are hard, right?
[19:40] Because then it is just the reciprocal sign goes in.
[19:42] So the trick is just change everything in terms of sine and cosine.
[19:45] So all trig functions can be changed in terms of sine and cosine. So this is the
[19:53] limit as x goes to 0, x times 1 over cosine of x times 1 over sine of x.
[20:11] Now what?
[20:15] What does this guy go to as x goes to zero?
[20:19] One.
[20:20] One.
[20:21] So if you want to split it up we can.
[20:28] The limit as x goes to zero of one over cosine times the limit as x goes to zero of x over
[20:37] sine of x.
[20:41] And then this direct substitution, we get 1.
[20:46] And this, we memorized as 1, the answer is 1.
[20:54] So the trick is just change all trig functions
[20:57] in terms of sine and cosine.

## 21:00 — Scaled-Angle Trigonometric Limits

[21:01] So here it's not sine of x over x, sine of x over 7x.
[21:11] So what can we do with that?
[21:17] Yeah, all we do is we factor this is,
[21:20] it's just a coefficient of like 1 7th.
[21:23] And we factor it out.
[21:38] We have 1 7 times 1 is 1 7.
[21:56] That one shouldn't have been too hard, but let's try this one.
[22:01] Can I factor out a 5 here?
[22:08] Definitely not.
[22:10] You can't factor a number out from inside a trigonometric function.
[22:14] Uh oh, so now what?
[22:17] Multiply by 5 over 5.
[22:20] Okay, I like that.
[22:24] Multiply this by 5 over 5.
[22:29] And take this guy out
[22:47] Why did you want that
[22:53] It functions.
[22:56] Yeah, so even though it's not sine of x over x,
[23:01] it's the same exact thing.
[23:06] And as x goes to 0, 5x also goes to 0.
[23:13] And this goes to 0 at exactly the same rate.
[23:17] If you wanted to, you could let like, uh, u, uh, yeah, let me go, u is five X.
[23:32] And the limit as X goes to zero of u.
[23:40] Okay.
[23:40] This also goes to zero and you end up with something like this.
[23:44] you don't really have to do this, but sine of u over u.
[23:52] And this limit, whatever the variable is, is equal to 1.
[24:03] So the key here, as long as whatever is inside the sine
[24:08] has to be that same thing down here.
[24:13] This will equal to, well, in this case, 5.
[24:18] This will equal, in this case, 5.
[24:24] bring the letters out.
[24:29] Oh, here's a good one.
[24:34] All right.
[24:34] New topic.
[24:35] Yeah.

## 24:37 — Absolute Value and One-Sided Limits

[24:38] We try direct substitution, we get 3 minus 3, or 3 minus 3, we get 0 over 0.
[24:48] What in the world are we going to do here?
[24:58] Uh, this absolute value, what kind of function is that?
[25:08] What's that? It's a piecewise function. Okay, let's look at the definition of the absolute value of x
[25:17] minus three. This is a piecewise function. It's either going to be x minus three or it's
[25:18] x minus three if x minus three is greater than or equal to zero and if x
[25:26] minus three is less than zero so x minus three it's either x minus three if x is
[25:41] Yeah, greater than or equal to zero.
[25:44] Sorry, greater than or equal to three.
[25:49] Negative x plus three.
[25:54] Let me just leave this x minus three.
[26:00] If x is less than three.
[26:03] Okay, all absolute value functions
[26:05] are piecewise functions, right?
[26:06] They all, they look like a V.
[26:08] There's two different pieces.
[26:15] Okay, so going back here, we're going to have to write a piecewise definition.
[26:27] So we're approaching three here, right?
[26:30] Not from left to right, but both sides.
[26:33] But this thing depends on each side of the three.
[26:36] So we're gonna have to break these up into the left-hand limit and the right-hand limit.
[26:43] So the limit as x approaches 3.
[26:54] Start from the left.
[27:06] OK, so since we're going from the left, are we always greater than or equal to 3 or less
[27:14] than 3?
[27:16] Less than 3.
[27:19] So I'm going to substitute this for this right here.
[27:23] So this is the limit as x approaches 3 from the left
[27:30] of negative, open parenthesis, x minus 3, close parenthesis, over 3 minus x.
[27:44] Which is the limit as x approaches 3 from the left.
[27:48] I distribute that negative and switch it, we get 3 minus x over 3 minus x, which simplifies
[27:59] to 1.
[28:04] And now let's take the limit as x approaches 3 but from the right-hand side.
[28:15] Now, since we're approaching three from the right, this absolute value.
[28:26] Okay, we're to the right of three, so I can substitute this in.
[28:33] So this is x minus three over three minus x.
[28:41] And...
[28:46] How do I simplify this?
[28:53] You can factor out the negative.
[28:57] Either from the numerator or denominator, we get negative x plus three, which if you
[29:09] around they simplify to 1 and now we get negative so the limit from the left
[29:19] exists it's equal to 1 the limit from the right exists negative 1 so what is
[29:26] the limit as X approaches 3 this does not exist
[29:46] Just to get an idea of this function right here, I think it looks something like this.
[29:54] everywhere to the right.
[30:00] It's got a cool function.
[30:04] We're to the right of 3, it's always going to be negative 1, or left to be positive 1.
[30:10] Okay, so what's this trick?
[30:12] Is we have to break up piecewise functions into two pieces.
[30:18] And especially involving absolute value.
[30:20] Okay, so break up.
[30:27] Okay, especially involving anything involving absolute values is a piecewise function.

## 30:47 — Piecewise Limits and Continuity

[30:48] All right, what are we going to do here?
[30:53] Actually, let's do a, no, let's just do this one.
[31:02] Here we cannot just use direct substitution.
[31:07] If I use direct substitution, you plug in 1 to the function.
[31:13] What is f of 1?
[31:18] Do I plug it into here or do I plug it into here?
[31:23] The top one, because that's in this domain right here.
[31:27] So that's ln of 1, which is equal to 0.
[31:34] We can't use direct substitution because, well we can only if we know it's continuous.
[31:41] And we don't know if this function is continuous because this has two different pieces
[31:46] Okay, so especially any time it changes right here at whatever number this is
[31:52] We got to just break up and do the left-hand limit and the right-hand limit. So let's start with the left
[31:59] X approaches
[32:01] 0 sorry 1
[32:04] From the left
[32:18] Now, since we're always to the left, what is f of x?
[32:27] Is x squared.
[32:28] So I can write the limit as x approaches 1 from the left of x squared.
[32:36] Now this is a continuous function.
[32:38] So we can just use direct substitution.
[32:42] If we plug it in, we get 1 squared, which is 1.
[32:50] As we approach from the
[32:56] right of f of x
[33:10] Okay, since we're always to the right of one, our function is equal to ln of x.
[33:18] So I can substitute in for f of x ln of x.
[33:25] And again, this is a continuous function, so we can just plug it in.
[33:33] And we get ln of 1 and ln of 1 is 0.
[33:44] So the left-hand limit exists, the right-hand limit exists.
[33:49] So what is this limit as x approaches 1?
[33:53] It does not exist because the left-hand limit does not equal the right-hand limit.
[34:07] A more interesting question is: what if the lower piece were x squared plus c?
[34:17] The question is, what value of C makes this continuous everywhere?
[34:29] everywhere.

## 34:30 — Choose a Constant for Continuity

[34:32] So you look at each piece.
[34:34] Is this continuous everywhere?
[34:38] Not really.
[34:40] It is continuous on what domain?
[34:42] Why won't it change colors?
[34:44] As long as x is greater than zero, it's continuous.
[34:48] And it's only this when it's continuous, or sorry,
[34:51] when it's that function.
[34:52] So this is continuous everywhere here.
[34:55] This is, no matter what c is, is a polynomial.
[34:59] So it's definitely continuous everywhere here.
[35:01] So the only question is, at 1, we got to make it continuous.
[35:07] So we got to, this is.
[35:21] This is natural log.
[35:24] And then this is just x squared.
[35:33] So we just have to shift it up or down so at one they're exactly the same value.
[35:40] Long story short, we just need to have the two limits equal.
[35:47] With plus c, the left-hand limit would be 1 plus c.
[35:57] So to make it continuous, we just
[35:58] have to make sure these match.
[36:02] So 1 plus C equals 0.
[36:04] C is negative 1.
[36:09] OK, that was, we're going to get to more
[36:11] about continuous functions later,
[36:13] but just a little quick question. Let's see.