WEBVTT



00:00:05.060 --> 00:00:06.800
Okay, today we're talking about...

00:00:07.940 --> 00:00:09.000
What are we talking about?

00:00:11.080 --> 00:00:14.280
We're going to have a quiz on Wednesday, not today.

00:00:15.160 --> 00:00:16.360
I want you to do this last thing.

00:00:19.940 --> 00:00:22.800
I want to put a limit as infinity.

00:00:37.220 --> 00:00:48.751
That means either the limit as x approaches infinity or some function, the limit as

00:00:48.751 --> 00:00:49.520
x

00:00:49.520 --> 00:00:51.360
approaches negative infinity.

00:00:52.880 --> 00:00:53.780
Some other way.

00:01:01.820 --> 00:01:05.280
This is the exact same thing we did in math 2.3,

00:01:05.700 --> 00:01:06.660
This is the same idea you studied as the end behavior of a function.

00:01:07.580 --> 00:01:10.280
It's the exact same thing as the end behavior of a function.

00:01:39.120 --> 00:01:41.700
Pretend that's some quadratic.

00:01:48.200 --> 00:01:56.240
The limit as x approaches infinity of x squared is infinity.

00:02:02.700 --> 00:02:09.611
So the limit as x goes to infinity, we want to know as x gets larger, what is f of x

00:02:09.611 --> 00:02:10.361
doing?

00:02:09.940 --> 00:02:12.340
And for quadratics, what's it doing?

00:02:13.660 --> 00:02:15.360
Getting larger and larger and larger.

00:02:15.940 --> 00:02:19.520
All polynomials either go up to positive infinity or go down to negative infinity.

00:02:20.680 --> 00:02:22.540
So our answer here would be infinity.

00:02:33.560 --> 00:02:36.880
Let's say some fourth degree polynomial.

00:02:38.700 --> 00:02:45.080
as x goes to infinity of whatever this function is.

00:02:48.400 --> 00:02:51.438
Again, if it's a polynomial, the answer is either going to be positive or negative

00:02:51.438 --> 00:02:52.188
infinity.

00:02:53.860 --> 00:02:58.960
So here, as x gets larger, this keeps going down and down and down and down.

00:02:59.400 --> 00:03:01.000
So this would be negative infinity.

00:03:01.000 --> 00:03:03.400
Okay.

00:03:20.360 --> 00:03:21.760
Oops.

00:03:21.760 --> 00:03:28.600
Let me do this one.

00:03:38.220 --> 00:03:45.760
Rational functions often had horizontal asymptotes.

00:03:46.760 --> 00:03:47.880
Do we remember this?

00:03:50.220 --> 00:04:04.480
And here, the limit, let's call this g of x, as x goes to infinity, this gets closer

00:04:04.480 --> 00:04:05.680
and closer to the number 3.

00:04:05.900 --> 00:04:11.020
It will never be 3, but this has a horizontal asymptote, the answer is 3.

00:04:17.100 --> 00:04:22.540
And then there's cases like sine of x.

00:04:28.120 --> 00:04:30.080
Let's say infinity or negative infinity.

00:04:32.580 --> 00:04:36.240
This is bounded, but will ever go approach a single number?

00:04:37.240 --> 00:04:39.340
No, so like this, something like this, the answer

00:04:39.340 --> 00:04:48.375
does not exist. So all the answers to limit as X goes either positive or negative

00:04:48.375 --> 00:04:49.125
infinity

00:04:48.940 --> 00:04:49.929
will either be positive infinity if it keeps going up forever, negative infinity if

00:04:49.929 --> 00:04:50.679
it

00:04:58.870 --> 00:05:06.530
examples. What kind of function is 4 over x? Rational.

00:05:07.810 --> 00:05:14.090
OK. You can kind of do some math with infinity.

00:05:14.870 --> 00:05:16.090
It's kind of like direct substitution.

00:05:17.650 --> 00:05:24.210
OK. 4 divided by a very, very large number is?

00:05:24.430 --> 00:05:27.390
Yeah. Well, yeah, I won't get there.

00:05:27.510 --> 00:05:30.950
Sorry. Let's say this is some number, very large number.

00:05:31.190 --> 00:05:35.390
4 divided by a very, very large number is a very, very small number.

00:05:35.390 --> 00:05:41.890
okay and the larger you plug in the smaller gets okay so in terms of limits

00:05:41.890 --> 00:05:48.730
The larger the magnitude of x becomes, the smaller the quotient becomes.

00:05:48.730 --> 00:06:02.210
zero and the same is X goes to negative infinity any number over negative

00:06:02.210 --> 00:06:07.510
infinity keeps getting smaller and smaller and smaller even though it's negative we

00:06:07.510 --> 00:06:17.410
don't care that is also zero. So when you try the direct substitution even when

00:06:17.410 --> 00:06:24.610
we're plugging in infinity anytime you have any number over infinity the limit

00:06:24.610 --> 00:06:25.630
It always goes to zero.

00:06:59.330 --> 00:07:00.770
Let me change this to minus.

00:07:02.890 --> 00:07:04.490
Sorry, let me change this one to minus.

00:07:09.310 --> 00:07:11.150
Here we got another rational function.

00:07:15.070 --> 00:07:25.450
If you try plugging in infinity, we get 3 times infinity squared minus infinity plus

00:07:25.450 --> 00:07:30.850
2, or 4 times infinity squared minus 5.

00:07:35.110 --> 00:07:36.650
What is infinity squared?

00:07:38.870 --> 00:07:57.051
3. Infinity times 3 is infinity. Infinity minus infinity plus some number 2. What

00:07:57.051 --> 00:07:58.350
are

00:07:58.350 --> 00:08:04.330
we doing here is a good question. You can still think of negative infinity plus 2 as

00:08:04.330 --> 00:08:14.190
negative infinity either way what is infinity minus infinity no we don't know

00:08:14.190 --> 00:08:21.670
that's an indeterminate form very important infinity minus infinity is an

00:08:21.670 --> 00:08:26.270
indeterminate form infinity plus infinity is not indeterminate what does

00:08:26.270 --> 00:08:31.970
that equal that's just infinity a large number plus a large number is a large

00:08:33.370 --> 00:08:39.922
But a large number minus another large number, we can't really do math with

00:08:39.922 --> 00:08:40.672
infinity.

00:08:41.290 --> 00:08:42.590
So that's an indeterminate form.

00:08:42.790 --> 00:08:44.850
So our point is we need to do something else here.

00:08:46.030 --> 00:08:47.310
And what are we going to do here?

00:08:54.210 --> 00:08:58.990
Some of you were probably taught last year just erase this stuff.

00:09:01.930 --> 00:09:03.870
Can we just erase stuff?

00:09:04.910 --> 00:09:05.810
No.

00:09:07.750 --> 00:09:09.170
The answer is sure and poor but why?

00:09:12.350 --> 00:09:14.550
Because Mr. Williams just said erase it.

00:09:14.770 --> 00:09:15.670
Erase it.

00:09:15.150 --> 00:09:17.230
I'm going to use the last 3.

00:09:24.250 --> 00:09:25.150
Oh, that was canceled.

00:09:26.630 --> 00:09:29.830
The x goes to infinity of 3 fourths. It's just 3 fourths.

00:09:31.450 --> 00:09:33.250
Okay, we just can't erase stuff.

00:09:35.970 --> 00:09:37.230
No, we can't.

00:09:39.750 --> 00:09:41.330
So what can we do?

00:09:44.050 --> 00:09:47.430
How can I show what's going to be equal to 3 fourths?

00:09:52.530 --> 00:09:54.870
You might be able to factor, I don't know if that's going to help.

00:09:55.550 --> 00:09:56.770
I don't even know if these factor.

00:09:57.190 --> 00:09:58.090
Yeah, I guess it does.

00:09:59.330 --> 00:10:00.230
That's not going to help.

00:10:02.430 --> 00:10:03.330
There's a trick.

00:10:05.150 --> 00:10:06.510
We should add it to our tricks.

00:10:06.510 --> 00:10:11.030
Let me copy this real quick.

00:10:27.130 --> 00:10:30.870
We're going to multiply the numerator and denominator by something.

00:10:30.870 --> 00:10:31.950
I'll do that.

00:10:35.410 --> 00:10:36.310
What?

00:10:41.210 --> 00:10:42.650
Which is just the number one,

00:10:42.870 --> 00:10:45.490
as long as X is not zero and it's definitely not zero.

00:10:46.050 --> 00:10:47.430
We're going to infinity large numbers.

00:10:49.990 --> 00:10:53.150
And I'm gonna choose the smaller of the two degrees.

00:10:55.090 --> 00:10:57.190
This is a second degree in the numerator,

00:10:57.350 --> 00:10:58.590
a second degree, they're the same degree.

00:10:58.590 --> 00:11:02.290
I'm going to choose 1 over x to the smaller degree, but they're the same.

00:11:03.150 --> 00:11:04.850
And then I distribute that.

00:11:06.270 --> 00:11:10.820
After multiplying by one over x squared, the expression becomes three minus one over

00:11:10.820 --> 00:11:15.370
x plus two over x squared, divided by four minus five over x squared.

00:11:16.230 --> 00:11:23.770
That over x plus 2 over x squared.

00:11:36.510 --> 00:11:37.410
OK.

00:11:38.030 --> 00:11:43.338
And then using our limit laws, the limit of a quotient is the quotient of limits,

00:11:43.338 --> 00:11:44.088
and

00:11:43.670 --> 00:11:46.370
then the limit of a sum is the sum of limits.

00:11:46.890 --> 00:11:50.717
We're not going to write all that, because it's just writing lots of different

00:11:50.717 --> 00:11:51.467
limits.

00:11:54.190 --> 00:11:56.590
But, let's just look at each single piece.

00:11:57.470 --> 00:12:00.250
As x goes to infinity, what does 1 over x go to?

00:12:02.110 --> 00:12:03.010
Zero.

00:12:03.990 --> 00:12:09.910
Two over infinity squared is two over infinity, which goes to zero.

00:12:12.170 --> 00:12:15.290
Five over infinity squared goes to zero.

00:12:18.130 --> 00:12:32.430
So the numerator goes to 3 minus 0 plus 0 and 4 minus 0, the answer is 3 fourths.

00:12:38.870 --> 00:12:45.290
When the degrees are equal, the limit is the ratio of the leading coefficients.

00:12:45.290 --> 00:12:49.450
taught right we look at the two degrees they're the same it's just the leading

00:12:49.450 --> 00:12:58.910
coefficient divided by the leading coefficient but this is why it would

00:12:58.910 --> 00:13:05.850
anything change if I went to negative infinity? No, the negative infinity that

00:13:05.850 --> 00:13:14.250
still goes to 0 0 0 we still get 3 4ths. Okay so this rational function has a

00:13:14.250 --> 00:13:18.450
horizontal asymptote I don't know what it looks like in between but definitely

00:13:18.450 --> 00:13:27.407
a 3 4ths goes here and here or something and it has some I don't know what it looks

00:13:27.407 --> 00:13:28.350
like in

00:13:28.350 --> 00:13:32.830
but something like that

00:13:55.230 --> 00:13:58.350
So here's another rational function.

00:14:08.550 --> 00:14:16.781
Right, this is a fifth degree polynomial as x goes to positive infinity, this goes

00:14:16.781 --> 00:14:17.531
to

00:14:17.330 --> 00:14:22.444
positive infinity, this is a third degree polynomial with a negative leading

00:14:22.444 --> 00:14:23.194
coefficient.

00:14:24.610 --> 00:14:26.530
It goes to negative infinity.

00:14:27.030 --> 00:14:28.650
And what is infinity divided by negative infinity?

00:14:30.570 --> 00:14:31.470
Indeterminate form.

00:14:31.330 --> 00:14:32.230
Don't know.

00:14:34.250 --> 00:14:35.150
OK.

00:14:34.990 --> 00:14:36.330
So, we've got to do some more stuff.

00:14:38.270 --> 00:14:39.590
So what are we going to multiply by?

00:14:40.050 --> 00:14:48.690
One over x cubed divided by one over x cubed.

00:14:52.150 --> 00:14:58.761
I'll show you if you don't choose the smaller of the two what can happen but we

00:14:58.761 --> 00:14:59.511
distribute

00:15:37.730 --> 00:15:46.438
As X goes to infinity, 5 over infinity goes to zero, 1 over infinity cubed goes to

00:15:46.438 --> 00:15:47.188
zero,

00:15:46.950 --> 00:15:51.910
negative seven over infinity cube goes to zero.

00:15:54.190 --> 00:15:55.090
Okay.

00:15:55.490 --> 00:15:57.890
What does infinity square go to?

00:16:01.370 --> 00:16:03.850
Over negative three.

00:16:05.170 --> 00:16:06.930
What is infinity divided by negative three?

00:16:08.910 --> 00:16:10.750
Negative infinity.

00:16:11.910 --> 00:16:13.970
A positive number, right?

00:16:13.970 --> 00:16:16.310
Infinity stands for a very, very large positive number

00:16:17.390 --> 00:16:19.710
divided by negative three

00:16:21.210 --> 00:16:23.850
is a negative some very large number.

00:16:24.050 --> 00:16:25.610
So it's a negative, negative, negative.

00:16:28.170 --> 00:16:33.770
Okay, I chose the smaller of the two degrees.

00:16:34.870 --> 00:16:37.590
If you didn't, you run into problems.

00:16:41.030 --> 00:16:43.090
Let me show you what happens if

00:16:54.690 --> 00:16:57.630
Let's say I chose x to the fifth.

00:17:00.470 --> 00:17:14.606
You would get 1 minus 5 over x cubed plus 1 over x cubed over negative 3 over x

00:17:14.606 --> 00:17:15.356
squared

00:17:15.350 --> 00:17:18.810
minus 7 over x cubed.

00:17:20.130 --> 00:17:22.850
You don't have to write this down.

00:17:23.710 --> 00:17:29.690
So x goes to infinity, this goes to zero, this goes to zero, this goes to zero, this

00:17:29.690 --> 00:17:43.592
goes to zero, and we get one minus zero, plus zero over zero, plus zero, one over

00:17:43.592 --> 00:17:44.410
zero.

00:17:47.170 --> 00:17:49.290
And what does a number over zero go to?

00:17:50.630 --> 00:17:52.290
In terms of limits.

00:17:54.210 --> 00:17:56.070
Either positive or negative infinity.

00:17:57.930 --> 00:18:02.842
But what we don't know here, since we did it here, we don't know, right, the

00:18:02.842 --> 00:18:04.070
denominator goes to zero.

00:18:05.110 --> 00:18:09.929
But we don't know if it's from the left side, which is a negative number, or from

00:18:09.929 --> 00:18:11.630
the right, it's a positive number.

00:18:12.210 --> 00:18:15.610
So we don't know what this ends up as positive or negative infinity.

00:18:18.010 --> 00:18:26.390
OK, so we don't want to do this. OK, so anyways, the point was,

00:18:27.770 --> 00:18:31.931
do one over the smaller of the two degrees and then it will show you whether it's

00:18:31.931 --> 00:18:32.910
positive or negative infinity.

00:18:32.910 --> 00:18:34.000
Let's try another one.

00:18:39.530 --> 00:18:43.610
Again, if you plug in infinity, you get infinity over infinity indeterminate form.

00:18:43.730 --> 00:18:44.790
So what am I going to multiply by?

00:18:47.990 --> 00:18:50.002
The numerator has degree one and the denominator has degree two, so multiply by one

00:18:50.002 --> 00:18:50.752
over x.

00:18:51.630 --> 00:18:53.470
So one over x, one over x.

00:18:53.570 --> 00:18:54.770
A smaller two degrees.

00:19:25.210 --> 00:19:28.650
this goes to 0, this goes to 0

00:19:30.750 --> 00:19:32.550
1 plus 0

00:19:33.490 --> 00:19:35.990
this goes to negative infinity

00:19:38.470 --> 00:19:39.790
minus 0

00:19:42.950 --> 00:19:45.410
what is 1 over negative infinity go to?

00:19:46.690 --> 00:19:49.570
any number over positive infinity goes to 0

00:19:57.490 --> 00:19:59.150
So these are three different cases.

00:19:59.370 --> 00:20:04.132
One here, when the degree in the denominator is greater, this always ends up going

00:20:04.132 --> 00:20:04.882
zero,

00:20:05.290 --> 00:20:07.450
but this is kind of why we showed it.

00:20:09.050 --> 00:20:14.081
Here when the degree in the numerator is larger, it's either going to go to positive

00:20:14.081 --> 00:20:14.831
or negative

00:20:14.710 --> 00:20:15.610
infinity.

00:20:15.070 --> 00:20:21.630
You have to use this to figure out which one and then when the degrees are the same

00:20:23.070 --> 00:20:26.290
It is always the leading coefficient divided by the leading coefficient

00:20:28.210 --> 00:20:30.430
And this was why it's the case

00:20:34.770 --> 00:20:40.730
Okay, so that was rational functions

00:21:02.450 --> 00:21:04.770
What type of function is this?

00:21:06.310 --> 00:21:07.210
Exponential.

00:21:09.130 --> 00:21:12.890
Okay, you can kind of do infinity math of this

00:21:14.530 --> 00:21:16.090
Three times infinity

00:21:18.410 --> 00:21:23.250
Three times infinity is still infinity. What is e to a very very large number?

00:21:28.450 --> 00:21:31.770
Like what's e to a thousand at which is not very large

00:21:33.660 --> 00:21:35.480
It's an enormous number.

00:21:36.640 --> 00:21:37.640
Big, big, big, big number.

00:21:38.060 --> 00:21:39.640
Okay, this is infinity.

00:21:47.940 --> 00:21:56.120
And then, let's see what happens as we go to negative infinity.

00:22:17.560 --> 00:22:22.540
What is e to a negative exponent?

00:22:31.880 --> 00:22:35.220
It is 1 over e to positive infinity.

00:22:36.780 --> 00:22:41.160
Okay, e to positive infinity is infinity.

00:22:42.620 --> 00:22:44.300
What is 1 over infinity?

00:22:45.240 --> 00:22:46.140
Zero.

00:22:45.600 --> 00:22:46.500
It goes to zero.

00:22:47.840 --> 00:22:48.740
Okay.

00:22:49.780 --> 00:22:55.100
All exponential functions are either the 3x,

00:22:55.200 --> 00:22:56.280
looks something like this.

00:22:57.340 --> 00:22:59.540
All exponential functions on one side,

00:22:59.620 --> 00:23:01.380
it goes to either positive or negative infinity.

00:23:01.900 --> 00:23:03.640
On the other side, we always have a horizontal asymptote

00:23:03.640 --> 00:23:04.540
of zero.

00:23:09.000 --> 00:23:12.020
So that's the end behavior for all exponential functions.

00:23:13.500 --> 00:23:16.720
If this were like negative 3x, it would just be flipped.

00:23:17.500 --> 00:23:18.864
this will go to 0 and this would go to infinity. So you do got to be careful to sign

00:23:18.864 --> 00:23:19.614
this up.

00:23:19.950 --> 00:23:21.330
And then we'll do the other case.

00:23:23.650 --> 00:23:29.850
For the radical expression, first evaluate the negative-infinity case.

00:23:34.670 --> 00:23:42.550
Nine times negative infinity squared plus negative infinity, well forget that.

00:23:43.290 --> 00:23:44.390
This is a quadratic.

00:23:45.890 --> 00:23:50.530
Where does a quadratic go as you go to negative infinity?

00:23:52.830 --> 00:23:54.410
It goes to positive infinity, right?

00:23:54.610 --> 00:23:56.390
That quadratic looks something like this.

00:23:58.290 --> 00:24:00.790
So the inside here goes to positive infinity.

00:24:02.330 --> 00:24:10.890
The square root of positive infinity is... square root of a very big number is still

00:24:10.890 --> 00:24:11.790
a really big number.

00:24:11.770 --> 00:24:12.710
So that goes to infinity.

00:24:12.710 --> 00:24:16.770
And then we got minus 3 times negative infinity.

00:24:19.310 --> 00:24:21.370
What does negative 3 times negative infinity go to?

00:24:22.090 --> 00:24:22.990
Infinity.

00:24:23.990 --> 00:24:26.290
And so our answer is infinity.

00:24:29.870 --> 00:24:31.710
This is infinity.

00:24:34.510 --> 00:24:39.210
Now the other one is much more interesting.

00:24:47.650 --> 00:24:54.370
Here if you plug in infinity, we get infinity minus infinity, which is what?

00:24:56.830 --> 00:25:00.590
In determinant, in terms of limits it's an indeterminate form.

00:25:02.210 --> 00:25:04.310
Ok so what are we going to do here?

00:25:08.690 --> 00:25:09.810
It's one of our old tricks.

00:25:12.170 --> 00:25:15.490
We're going to multiply by the conjugate, thank you whoever said that.

00:25:17.430 --> 00:25:29.490
Okay, so we're going to go the square root of 9x squared plus x plus 3x.

00:25:46.890 --> 00:26:12.768
plus x. The 2 milli terms cancel out, we get minus 9x squared. And now what happens?

00:26:12.768 --> 00:26:14.290
These

00:26:16.690 --> 00:26:18.090
cancel.

00:26:24.290 --> 00:26:27.330
x over x squared.

00:26:33.970 --> 00:26:37.410
Which we plug infinity, we still get infinity over infinity.

00:26:46.910 --> 00:26:52.591
We're either going to multiply by 1 over x, or we're going to factor out an x out of

00:26:52.591 --> 00:26:53.341
here.

00:26:53.930 --> 00:26:56.430
Let's do this one.

00:27:04.370 --> 00:27:07.730
We're running out of time, so I've got to speed this up.

00:27:19.390 --> 00:27:22.011
After rationalizing, divide by x and move the positive x into the square root as the

00:27:22.011 --> 00:27:22.830
square root of x squared.

00:27:26.310 --> 00:27:29.290
Is x equal to the square root of x squared?

00:27:34.870 --> 00:27:37.010
This is very important.

00:27:38.990 --> 00:27:42.357
The identity x equals the square root of x squared is valid here because x

00:27:42.357 --> 00:27:43.107
approaches positive infinity.

00:27:46.990 --> 00:27:50.030
If x is a negative number this is not true.

00:27:50.770 --> 00:27:58.930
In fact, if x is a negative number, we have to put a negative here.

00:28:02.230 --> 00:28:03.130
Oh, it's about to ring.

00:28:03.350 --> 00:28:04.250
So which one is it?

00:28:04.210 --> 00:28:05.330
Is it a positive or a negative?

00:28:05.410 --> 00:28:06.790
Which one am I going to substitute in?

00:28:08.210 --> 00:28:13.269
Since we're going to positive infinity, I'm going to substitute this x right here

00:28:13.269 --> 00:28:14.019
for

00:28:13.630 --> 00:28:20.250
this, we get the limit as x goes to infinity.

00:28:22.250 --> 00:28:28.290
Get rad nine x squared plus x over rad x squared,

00:28:30.430 --> 00:28:33.830
which now I can make one big square root.

00:28:41.170 --> 00:28:50.730
We get the square root of 9 plus 1 over x plus 3.

00:28:51.410 --> 00:28:52.730
Hold on one sec.

00:28:54.830 --> 00:28:58.910
Now as x goes to infinity this little guy goes to zero.

00:28:58.910 --> 00:29:03.470
As x approaches infinity, one over x tends to zero, so the limit is one sixth.

00:29:05.470 --> 00:29:08.370
Okay, be real careful when you bring in

00:29:09.070 --> 00:29:10.510
into square roots.

00:29:11.470 --> 00:29:13.010
Sometimes this, sometimes that.

00:29:13.950 --> 00:29:14.850
Okay, the homework is some delta math.