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For more information, please see full course syllabus of Math Analysis
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Lecture Comments (4)

1 answer

Last reply by: Professor Selhorst-Jones
Sun Oct 5, 2014 11:32 AM

Post by Nico Han on October 1, 2014

me too! Math suddenly become so interesting to me. I was getting 85, but now I am getting 98. Thanks a bunch!

1 answer

Last reply by: Professor Selhorst-Jones
Tue Nov 19, 2013 5:41 PM

Post by Tim Zhang on November 19, 2013

such a great teacher!!, I used to got so confused on the math theory my high school told me, because she never told me the thinking process to get these conclusion. Now, I got 100 on my pre-cal all the time, haha. than you so much!

Horizontal Asymptotes

  • A horizontal asymptote is a way of asking what happens to a rational function in the "long run". Is there a vertical location which the function approaches as the horizontal location "slides to infinity"? Symbolically, we can express a horizontal asymptote as
    x → ±∞       ⇒        f(x) → b.
    Informally, a horizontal asymptote is a vertical height that the function is "pulled" towards as it moves very far left or right.
  • Not all rational functions have horizontal asymptotes.
  • To find a horizontal asymptote, expand the polynomials of the numerator and denominator so you have something in the form
    f(x) =an xn + an−1 xn−1 + …+ a1 x + a0

    bm xm + bm−1 xm−1 + …+ b1 x + b0

    Notice that n is the numerator's degree and m is the denominator's degree. There are three possibilities:
    • If n < m, then there is a horizontal asymptote at y=0 (the x-axis).
    • If n=m, then there is a horizontal asymptote based on the ratio of the leading coefficients:
      y =an


    • If n > m, there is no horizontal asymptote.
  • A slant asymptote is similar to a horizontal asymptote, but instead of approaching a horizontal line, it approaches some slanted line in the "long run". It doesn't settle down to a single value, but it does get "pulled" along the slanted line as it moves very far left or right.
  • A rational function has a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator (n = m+1 from the above). We find the asymptote by polynomial division. Break the function into two parts: a portion with no denominator (the asymptote) and the remainder to the division still over the denominator (which goes to 0 in long term). [This method also works to find horizontal asymptotes.]
  • Graphically, we represent horizontal and slant asymptotes the same way we did vertical asymptotes: with a dashed line. However, unlike vertical asymptotes, the graph can cross over a horizontal or slant asymptote. Furthermore, there is only ever one horizontal/slant asymptote.

Horizontal Asymptotes

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:05
  • Investigating a Fundamental Function 0:53
    • What Happens as x Grows Large
    • Different View
  • Idea of a Horizontal Asymptote 1:36
  • What's Going On? 2:24
    • What Happens as x Grows to a Large Negative Number
    • What Happens as x Grows to a Large Number
    • Dividing by Very Large Numbers Results in Very Small Numbers
    • Example Function
  • Definition of a Vertical Asymptote 8:09
  • Expanding the Idea 9:03
  • What's Going On? 9:48
    • What Happens to the Function in the Long Run?
    • Rewriting the Function
  • Definition of a Slant Asymptote 12:09
    • Symbolical Definition
    • Informal Definition
  • Beyond Slant Asymptotes 13:03
  • Not Going Beyond Slant Asymptotes 14:39
  • Horizontal/Slant Asymptotes and Graphs 15:43
  • How to Find Horizontal and Slant Asymptotes 16:52
  • How to Find Horizontal Asymptotes 17:12
    • Expand the Given Polynomials
    • Compare the Degrees of the Numerator and Denominator
  • How to Find Slant Asymptotes 20:05
    • Slant Asymptotes Exist When n+m=1
    • Use Polynomial Division
  • Example 1 24:32
  • Example 2 25:53
  • Example 3 26:55
  • Example 4 29:22

Transcription: Horizontal Asymptotes

Hi--welcome back to Educator.com.0000

Today, we are going to talk about horizontal asymptotes.0002

In the previous lesson, we learned about the idea of a vertical asymptote,0005

a horizontal location where the function blows out to infinity, either up or down.0008

Symbolically, we can express this as a vertical asymptote as x approaches some horizontal location, a;0012

and when that happens, f(x) goes to positive or negative infinity.0020

We can flip this idea to the reverse and discuss the idea of a horizontal asymptote,0025

a vertical location which is approached as the horizontal location slides to infinity.0030

As our x becomes very, very large, what vertical height do we go to?0035

Symbolically, we can express it as x goes to positive or negative infinity (as in, x becomes very, very large,0039

either positively or negatively), and f(x) goes to some b, goes to some specific height y = b.0045

To understand this, let's take a look at our old friend from last time, our fundamental function, 1/x.0054

Notice that, as x grows large, we see f(x) shrink down very small; as we go far out, it becomes very, very small.0060

We can see that we are at 1/10 over here and -1/10 over here.0067

We can expand this to an even larger viewing window, and we can get a sense for just how small f(x) eventually becomes.0072

With our y going only from -0.5 to +0.5, we can see that, by the time we have made it to 100, we are at these tiny numbers.0078

We are 1/100 and -1/100, respectively; so we see that becomes really, really, really small, given enough time.0085

So, the farther we go out, this f(x) is going to sort of crush down to 0.0092

We can see this behavior, being sucked towards a certain height, in many rational functions.0098

In 5x/(x - 2), we see that it has this horizontal asymptote at 5; it sort of gets pulled towards a height of 5 in the long run.0102

Over here, with g(x) = (-3x2 + 6)/(x2 + 1), it gets pulled towards a height of positive 3.0112

In fact, it gets pulled really, really quickly.0120

Just like we had with vertical asymptotes, where it never quite touches the asymptote,0122

with a horizontal asymptote, it will not quite actually get to there.0127

It is going to get very, very close to it; and we will see that as we explore why this is occurring in just a few moments.0131

So, we will formally define this behavior in a little bit; and we will name it a horizontal asymptote.0137

But first, let's understand why it occurs: what is actually making this happen?0141

Let's start by investigating f(x) = 1/x: since x is in the denominator, as it grows really, really, really large,0146

there is a giant denominator that crushes the numerator.0152

The numerator just stays still--it just stays at 1; but the denominator gets big--it has x, and so it is able to march out forever.0156

So, as it gets really, really big, it crushes the numerator down to 0 in the long run.0163

So, if we look at the negative side...over here is negative; we have our vertical asymptote at 0,0168

and we are looking at what happens as it slides to the left.0176

We have -0.5, and it is at -2; -1, then -1; but as the numbers get larger and larger...at -5, we are at -0.2; at -10, we are at -0.1.0179

At -1000, we have made it to -.001; and it is just going to keep getting smaller and smaller and smaller.0188

Negative one billion will be a very, very tiny number.0195

Now, notice that there is no number we could plug in to actually get 0; we are just going to get very, very small numbers.0197

These giant denominators are going to make very, very small numbers that will approach 0.0202

We won't actually make it to 0, but we will get really, really, really close to it.0206

The same thing happens on the positive side, if we look at what happens as we go positive.0211

We start looking from our 0, and we go to the right; at 0.5, we are at 2; at 1, we are at 1.0215

At 5, we are at 0.2, and so on; we get that, at 1000, we are at .001.0222

And we as we get really, really large numbers, we will get crushed smaller and smaller and smaller.0227

Since the denominator grows so much faster than the numerator (the numerator isn't moving at all,0232

so it is not even growing at all), the fraction will eventually shrink to 0, as we get very large denominators crushing our numerator.0237

For any rational function, if the denominator's degree is greater than the numerator's degree--0243

that is, if the denominator is able to grow faster than the numerator is able to grow--the rational function will eventually go to 0.0248

If we have x2/(x3 + 5), that is eventually going to get crushed down to 0,0254

because x2 doesn't grow as fast as x3 + 5.0261

So, the x3 + 5 can effectively outrun x2 in the long run, so it will get much larger than x2 will.0265

And so, it will crush the whole thing down to 0.0271

So, if the degree is greater in the denominator (3 versus the numerator's degree, 2), it will eventually get crushed to 0.0273

How do we get rational functions with a horizontal asymptote that isn't at 0, then?0282

Let's look at one: f(x) = 5x/(x - 2); and of the two graphs that we saw, that was the one on the left, the red one that had a horizontal asymptote at 5.0287

If we plug in 1 (once again, this will be the negative side), what happens as we go more and more negative?0295

If we plug in +1, we will get 5(1)/(1 - 2), so we get 5/-1, which is -5.0302

If we plug in -10, we will get -50/-12, approximately 4.17.0314

-100 gives -500/-102, which is approximately 4.90; -1000 gives -5000/-1002, which is 4.99, approximately.0319

So, notice: as we plug in these things, the 5x here and the positive x here,0328

they end up growing at the same rate, other than this multiplicative factor of 5.0334

So, the top grows 5 times faster, precisely 5 times faster, than the bottom does.0339

So, as they go out one way or the other, they are going to end up approaching...the top is growing0344

5 times faster than the bottom is growing, so it is going to end up approaching 5,0350

because when we have very, very large numbers that we are going to put in (eventually, like 1 million),0354

it will be 5(really big number), divided by (really big number); so it will cancel out to 5.0358

We have this other factor of the -2 here; but as the numbers get much, much larger, like 1 million minus 2...1 million hardly notices the -2.0364

It has a slight effect, but it is not much of an effect.0373

And so, as we get down to farther and farther values out, as we get to larger and larger values,0376

it will have less and less of a relative effect, and we are going to get closer and closer to 5.0381

The exact same thing happens if we look at what happens on the positive side.0386

If we start by plugging in 3, we are at 15/1, so we are very different at 3; we are at 15.0390

But when we plug in 10, we are at 50/8; at 100, 500/98; at 1000, 5000/998.0396

We have this difference that becomes less and less impressive; this -2 becomes less and less meaningful.0402

And eventually, it becomes 5(number)/(number), which goes to 5; look at how close we have managed to make it by the time we are at 1000.0408

And this little pattern will just get closer and closer to 5 as we continue this pattern out; we will just get much closer to 5.0417

Once again, we will never touch 5, because we will always be off by this factor of -2; we will always be slightly imperfectly equal to 5.0424

So, it won't ever actually equal that horizontal asymptote precisely.0432

But it is going to get arbitrarily close to it; it is going to get really, really, really close, until we are dealing with numbers like 5.00001.0435

And so on, and so on...we will eventually be able to get to any arbitrary closeness we want, as long as we look at an x large enough.0442

Once we get far enough from the vertical asymptote at x = 2, we see that the numerator and denominator grow at a constant ratio, 5x and x.0449

So, for any rational function, if the degrees of the numerator and the denominator are equal,0456

we will get a horizontal asymptote that isn't equal to 0; we will get a horizontal asymptote at some height.0461

Notice: the 5x here and the x here have the same degree, a degree of 1.0466

So, since they have the same degree on the top and the bottom, they are going at the same rate, in a way.0471

Other than that multiplicative factor of 5, they are running in the same scale of growth.0476

So, since they have the same scale of growth, they are going to grow around the same rate,0481

which means it is only that multiplicative factor that is going to determine the height that it ends up at.0485

A horizontal asymptote is a horizontal line y = b, where, as x becomes very large (positive or negative), f(x) gets arbitrarily close to b.0490

Symbolically, we show this as "x goes to negative infinity" or "x goes to positive infinity"0505

means that f(x) will go to b; we are going to get to this height; we are going to move toward this height, surely, steadily.0510

It might not get perfectly to b; in fact, it almost certainly won't, as what we were just talking about.0520

But it will get really, really close to b; it will get arbitrarily close to it.0524

Informally, we can think of a horizontal asymptote as a vertical height that the function is pulled towards, as it moves very far left or right.0527

Over the long term, it will start somewhere else, but it gets pulled, in the long term, to a certain height,0535

until it gets really, really close to this horizontal asymptote.0540

We can take this idea and go beyond just having a horizontal asymptote.0544

Consider f(x) = (x3 - 1)/x2; as x gets large, we see f(x) grow very close to the line y = x.0548

We can see that on that dashed orange line going through the y = x line.0557

We see how close it becomes; in fact, it grows so close, so quickly, that we almost can't tell the difference between the two on the far parts in this graph.0561

It will never be perfectly the same, because we have this -1 here; but it will become really, really close to it.0569

This idea is similar to a horizontal asymptote, but it is no longer horizontal.0577

Since it is at a slant, we can't call it a horizontal asymptote; so instead, we call it a slant asymptote.0581

Sometimes, it is also called an oblique asymptote.0586

So, what is going on--why do we see this?0589

Once again, we are trying to consider what happens to the function in the long run.0591

The idea of all this horizontal asymptote/slant asymptote stuff is a question of what happens to this function0594

as we look at very large x--as we go really far right/as we go really far left.0600

In this case, we could plug in large numbers to see what happens; but that will slightly obscure some details for other slant asymptotes.0604

So, instead, what we want to notice is that we can rewrite the function.0612

If we have (x3 - 1)/x2, we could say, "Look, we can divide out the x3,0616

and we can break our fraction apart so that we get x3/x2 - 1/x2."0623

And so, the x3 and x2 cancel down to just x - 1/x2.0627

So, by using division, we can see this function in a new way.0632

In this form, it is clear that, as x goes to positive or negative infinity, this part right here,0636

since it is 1/x2, is just going to sort of get crushed down to 0 by its much larger denominator.0641

But the x here will end up just continuing on; it will just keep moving forward.0646

So, in the long run, as we get to very large x's, this part here goes to 0; but x continues going out.0651

So, effectively, the function will become just x.0658

We can also get this (x3 - 1)/x2...we can break it into this format0662

through the process of polynomial long division, which will be necessary0666

when we have slightly more complicated denominators that we are dealing with.0668

So, you write this as this, minus 1; so in that way, we would have x2.0674

How many times does x2 go into x3?0683

It goes in just x; x times x2 gets us x3.0684

We have nothing else, so we subtract by x3; that gets us 0.0688

We bring everything down; and so we have 0x2 + 0x - 1, which leaves us with just a remainder of -1.0692

We have this remainder of -1; so (x3 - 1)/x2 is equal to x, plus the remainder.0702

The remainder is -1; and then we put it back over our original denominator, the thing that we divided by.0710

We get x plus -1 over x2, which is the exact same thing that we had down here on this line.0717

We can get this through polynomial division, which will be necessary when we are dealing with slightly more complicated denominators.0724

We can also define a slant asymptote similarly to how we defined a horizontal one.0730

It is also called an oblique asymptote sometimes.0734

A slant asymptote is a line y = mx + b (remember, mx + b is just our normal slope-intercept form for a line),0737

where, as x becomes very large (positive or negative), f(x) gets arbitrarily close to that line, y = mx + b.0744

Symbolically, we can show this as: as x approaches negative infinity (very large negative values),0751

or x approaches positive infinity (very large positive values), f(x) will go to mx + b.0755

f(x) will approach just being the same as this line, mx + b.0761

Informally, a slant asymptote is a non-horizontal line that the function is pulled towards as it moves very far to the left or to the right.0765

So, we might start at different heights; but as we get farther and farther to x, we end up getting pulled along this line, this slant asymptote.0774

We can even go beyond the idea of a slant asymptote.0784

Really, the question we have been working on can be phrased as "What does the function look like in the long term?"0786

What is the long-term behavior of this function?0792

So far, we have answered that with horizontal and slant asymptotes.0795

But a function could also tend to a curve; it could tend to anything, really.0798

If we had (x4 + 17x + 20)/(10x2 - 10x - 20), well, notice:0802

we could make that as squared and to the fourth up here.0807

So, the degree of the top is one, two steps larger than the degree on the bottom.0812

That means that, over time, it is going to effectively be the same as x2 coming out of that, x4 divided by x2.0818

We are going to get something that looks kind of like an x2, which is a parabola, which is exactly what we see here.0826

As we get to very large x-values, we see it get pulled along this curve.0832

Notice this curve that we have of a parabola through here.0837

Now, it will behave differently when we are at the vertical asymptotes, because we have vertical asymptotes x - 2 and x + 1.0841

So, there are vertical asymptotes at -1 and vertical asymptotes at +2.0847

We will get pulled into these vertical asymptotes in various ways.0853

But in the long run, as it gets to very large x-values, it gets pulled into this parabolic shape.0857

In fact, if we were to divide this out through dividing the bottom into the top through polynomial division,0863

we would be able to find that it eventually is approaching the parabola 1/10 times x2 plus x plus 3.0870

And that is why we see that parabolic curve right there--pretty cool.0876

That said, we are not going to really see this in this course, or probably in any other course that you are taking right now.0880

While this shows us an interesting idea, don't expect to see this in a normal class.0885

Few textbooks and very few teachers will discuss anything beyond the idea of a slant asymptote.0888

As such, we will not be exploring the idea any further in this class, either.0893

However, it is useful to notice how all of these ideas have been linked.0896

They are about answering: "Where does this go in the long term? What is happening eventually to my function?0899

How will it behave when I look at very large values being plugged in?"0906

We can get a sense of this by thinking about what happens to the function as the numbers get larger and larger and larger.0910

What will happen--how, in what general way, will this function behave when we are plugging in x that is a million, a billion, a trillion--a really, really big x?0915

That is what all of these ideas in this lesson have been about.0924

What happens as x becomes very, very large--how does this thing behave?0927

It could behave in these non-slant asymptote things where it pulls into a parabola or some other polynomial shape.0931

But we are going to restrict ourselves to just horizontal and slant asymptotes, since that is what most other courses look at.0936

And they are also the easiest for us to approach right now.0941

Horizontal/slant asymptotes and graphs: Just like their vertical cousins, it is customary to show horizontal/slant asymptotes with a dashed line.0945

We normally use a dashed line to show, "Look, here is an asymptote."0953

So, if we had (4x4 + 3x3 + 10x2)/(2x4 + 1), we would get this graph over here on the right.0957

And notice: it has a horizontal asymptote at 2, and so, over the long run, our graph gets pulled towards this.0964

Now, notice that, in the middle, it has a behavior that is totally different.0972

It has this interesting behavior in the middle.0976

Unlike vertical asymptotes, the graph can cross the horizontal asymptote.0978

It is allowed to actually cross over that horizontal or slant asymptote.0983

Furthermore, there is only ever one horizontal or slant asymptote.0989

You can't have multiple horizontal/slant asymptotes, in the same way you can't have a vertical asymptote.0993

In any case, over the long run, the graph will be pulled along the asymptote.0998

That is the idea of an asymptote to really get across: that an asymptote is about the function eventually being pulled along it,1002

or, if it is a vertical asymptote, being stretched up along it vertically.1008

How to find horizontal and slant asymptotes: a horizontal or slant asymptote tells us how a function behaves in the long run; that is the idea here.1013

It is fairly easy to determine if a function has a horizontal asymptote, and, if so, what it is.1021

We will see a method for that first.1025

Finding a slant asymptote is a little bit trickier, though, and we will look at its method second; but it is not that difficult.1027

Any rational function is in the form n(x)/d(x), where they are both polynomials.1033

We can expand these polynomials into their normal form, our _xn + _xn - 1 + _xn - 2,1038

all the way until we eventually hit a constant; and the bottom one will be the same thing--some other blank.1047

So, an will be the blank on the top; bn will be the blank on the bottom.1052

And we have two different things; n will be the numerator's degree, and m will illustrate the denominator's degree.1056

There are going to be three possibilities: first, if n is less than m, then there is a horizontal asymptote at y = 0.1067

Why is that the case? Well, if n is less than m, then that means we have a big denominator, compared to the numerator.1075

The numerator will have a smaller degree; so it is going to grow slower,1083

which means that the bottom one will eventually grow large enough to crush the numerator.1087

So, we are going to crush it down to y = 0.1091

So, if we have a numerator degree that is less than the degree of the denominator,1094

the denominator will eventually grow large enough to crush the whole fraction down to 0.1100

If n is equal to m--if they are the same degree--if we have the same degree--then what we are going to see is:1105

we will see a horizontal asymptote that is based on the ratio of the leading coefficients.1114

We will get a horizontal asymptote, but it will no longer be set at y = 0.1118

Since xn and xm are basically the things that, in the long run,1121

are really going to determine how these polynomials are, and n = m,1125

then ultimately it is going to be anxn/bmxm,1129

in the long run, since xn and xm are the same value.1133

Since those things are at the front, they are going to really determine how the polynomial works in the long run.1137

And they will just cancel each other out, because we will have anxn/b...1142

and I will call it n as well, since we have n equal to m...xn.1147

Well, in the long run, what is effectively going to happen is that we will see these two things cancel each other.1151

And we will be left with just the ratio of the leading coefficients.1155

That will determine what the horizontal asymptote is going to go to--this ratio.1161

What is the leading coefficient on the top, divided by the leading coefficient on the bottom?1166

What is our first coefficient here and our first coefficient there?1170

Finally, if n is greater than m, if we have the numerator degree bigger, then there is no horizontal asymptote,1173

because the numerator is able to run faster than the denominator.1185

And so, it is able to escape the clutches of the denominator and actually keep going on to growing forever, and getting less and less.1189

It depends on how it is set up, specifically; but we will be able to have freedom on both the right and the left side,1195

as it manages to have very large values, because it will be able to outrun the denominator, because it has a larger degree.1200

All right, so let's talk about how to find slant asymptotes.1205

A rational function has a slant asymptote if the degree of the numerator is exactly 1 greater than the degree of the denominator.1208

So, in the terms we were using before, where n was the numerator's degree, and m was the denominator's degree, it would be n = m + 1.1215

We can find the asymptote through polynomial division.1222

For example, if we have (3x3 - 2x2 + 7x + 8)/(x2 - 3x), we say,1225

"Oh, look: there is a 2 in the denominator; there is a 3 on the numerator; is 3 = 2 + 1, the same thing as what is going on up here."1230

So, we are exactly 1 greater in the degree of the numerator than we are in the denominator.1240

So, we are going to have a slant asymptote.1247

At that point, we use polynomial division; so let's see how polynomial division would work here.1250

We have x2 - 3x as dividing into...what is in our numerator? 3x3 - 2x2 + 7x + 8.1254

So, how many times does x2 go into 3x3?1268

It is going to go in 3x; 3x times x2 gets us 3x3, so yes, we were right.1271

3x times -3x gets us -9x2; then we subtract this whole thing, so let's distribute that negative:1277

minus 3x2...that becomes + 9x2; so 3x3 - 3x3 is 0.1284

-2x2 + 9x2 is positive 7x2.1291

The next step: bring down the 7x, so + 7x; how many times does x2 go into 7x2? It goes in + 7 times.1295

So, 7x2...that checks out; 7 times -3 is -21x.1305

We subtract that whole thing and distribute our negative: minus, plus...7x2 - 7x2 is 0; 7x + 21x is 28x.1311

Bring down the 8; we get + 8 here; at this point, we see 28x--how many times can x2 go into 28x?1321

It can't go in anymore because of our degrees; so we have a remainder of 28x + 8.1330

And so, notice how these are the same thing: 3x + 7 is what we got as the result, 3x + 7 here.1337

And then, plus our remainder, 28x + 8, divided by what we started doing our division with...1344

that is how we get polynomial division; and indeed, 3x + 7 + (28x + 8)1353

divided by (x2 - 3x) is what our initial rational function is equal to; so we can break it down.1358

Break the function into two parts: a portion with no denominator (the portion with no denominator is 3x + 7--it doesn't have a denominator).1368

So, that is what our slant asymptote is, because 3x + 7 describes a line that is in the form mx + b.1377

So, 3x + 7 is a line, and the remainder to our division is our 28x + 8; it goes over the denominator x2 - 3x,1388

back over the denominator that we started with; and notice, (28x + 8)/(x2 - 3x), because it is the remainder...1404

the remainder is always going to have a degree less than what we started dividing with.1410

So, we have 28x1 divided by (x2 - 3x); so we have a bigger degree on the bottom than we do on the top.1414

In the long run, the denominator is going to crush the numerator; and this whole thing will go to 0, and we will be left with just 3x + 7.1421

So, in the long run, we will end up having a portion that has no denominator, and the portion that has a denominator,1429

because the degree on the bottom is now less after polynomial division--it will go to 0 in the long term.1435

Now, this method of polynomial division will also work to find horizontal asymptotes.1441

So, you can also use this method if you want to find the horizontal asymptote.1446

It is just that, instead of getting a line here, you will just get a constant--it will just be a constant value,1449

if it is a horizontal asymptote, as opposed to a slant asymptote.1458

So, you can also use polynomial division to find horizontal asymptotes.1463

But we have that other method for finding horizontal asymptotes that was pretty fast.1466

So, it is normally easiest to just use polynomial division when you want slant asymptotes.1469

All right, let's go over some examples.1474

f(x) = (10x5 + 3x4 + 8x2 - 2)/(2x5 + 27x3 + 12x).1475

Is there a horizontal or a slant asymptote?1483

What we do is compare the degree on the top to the degree on the bottom.1485

They are the same degree; so if they are the same degree, we have a horizontal asymptote.1489

Now, if we want to figure out what the horizontal asymptote is, well, we will figure that out.1497

And we just look at its ratio of leading coefficients; what is the leading coefficient for the top and the bottom?1502

The top has a leading coefficient of 10; the bottom has a leading coefficient of 2; we simplify that, and we get 5.1511

So, 5 is what our horizontal asymptote will be; so y = 5 is the horizontal asymptote.1521

Great; once we see that the degree on the top and the degree on the bottom are the same,1531

we know we have a horizontal asymptote that is not just going to be a 0.1539

And now, we figure it out by looking at the ratio of the leading coefficients,1542

because ultimately, the ratio of the leading coefficients on our biggest exponent, x's,1545

is going to be what determines what happens to these functions in the long run.1550

The next one: Using the graph of the function, determine a.1554

We have (12x3 + 5x2 - 10x + 8)/(ax3 + 2x - 2).1558

Now, we notice that there are 3 and 3; and here is a horizontal asymptote.1564

We have a horizontal asymptote, because the degrees are the same.1571

Also, we can see that we have a horizontal asymptote in our graph,1575

so it had better be the fact that the degree on the numerator and the denominator is the same.1579

At this point, we know...what is our horizontal asymptote? It is y = 3.1582

We see that by looking at where it goes; it cuts evenly between the 2 and the 4, so it must be at y = 3.1587

So, if that is the case, we know that the ratio of the leading coefficients, 12/a1592

(the leading coefficient on top is 12; on the bottom, it is a), must be equal to 3, our horizontal asymptote.1598

We work this out: 12 = 3a; divide by 3, and we get 4 = a, so a is 4; great.1607

We have (28x3 + 110x2 - 47x + 55)/(0.2x4 - 5x2 + 4).1616

Is there a horizontal or a slant asymptote? What is it?1624

In this case, the first thing, as always: we look to see what are our degrees.1627

The numerator degree is 3; the denominator degree is a 4.1631

So, if the denominator degree is larger, it crushes everything.1635

It is going to be able to eventually, in the long run, overtake the numerator.1639

It will run faster than the numerator and grow larger, and it will crush everything to 0.1643

So, there is a horizontal asymptote, but it is going to be the most boring horizontal asymptote of all,1647

but at the same time, kind of interesting: y = 0.1652

The denominator crushes that puny numerator; that numerator is just too small,1657

so the denominator crushes the puny numerator, because it has a larger degree.1663

And I want to point out that it seems at first...well, we have .2x4 and 28 times x3;1668

and there are also...110 and these other big numbers up here on the top.1674

But on the bottom, it all seems like pretty small, insignificant numbers.1678

So, why is it that the bottom is going to be bigger?...because it has a larger degree.1681

Ultimately, how a polynomial behaves in the long run is really determined by that degree.1686

The coefficients have effects; they affect things; but being really dominant is just determined by having the biggest degree.1691

x4, no matter how small that coefficient at the front is,1699

will eventually be able to outrun x3, no matter how big its coefficient is at the front.1702

So, that x4 will crush the numerator, because the numerator only has a degree of 3.1707

All right, the final example: at this point, we have (-4x4 + 7x3 + 23x2 - 43x + 5)/(x3 - 5x).1712

We are asked, "Is there a horizontal or slant asymptote?" and then, "What is it?"1720

The first thing we do is look at our degrees: we have 4 on the numerator and 3 on the denominator.1723

So, that means that the degree of the numerator is exactly 1 greater than the denominator.1728

If it is 1 greater, then that means we have a slant asymptote; and that makes sense,1741

because one degree larger than something else...if we divide them out...x50/x49...1747

that is going to become something along the lines of x.1753

So, it is going to have a nice linear form; it is going to become a line.1755

Degree 1 is a linear function, so that is why we see a slant asymptote, a line, coming out of it.1759

How do we figure it out? We figure it out by using polynomial division.1765

So, (x3 - 5x)...and notice: that is 5x1, not 5x2;1769

that is going to have a slight effect when we are doing our division.1775

-4x4 + 7x3 + 23x2 - 43x + 5.1779

x3: how many times does it go into -4x4?1791

It is going to go in -4x times, because -4x times x3 gets us -4x4.1793

-4x times -5x gets us 20x2; but the next thing we have is 7x3, so that is not going to go on the 7x3.1800

It is going to go on the 23x2 column.1808

-4x times -5x gets us 20x2; so 20x2 lines up there.1812

And it is positive; so at this point, we subtract by this amount.1818

So, we distribute our negative; that becomes positive; that becomes negative.1821

-4x4 + 4x4 becomes 0 (what it should be when we are doing polynomial division--the first part should always cancel to 0).1825

23x2 - 20x2 gets us positive 3x2.1831

The next step: we bring down the other things that we will end up using.1836

We bring down the 7x3; we bring down the -43x.1841

How many times does x3 go into 7x3? It will go in 7 times.1848

7 times x3 gets us 7x3; 7 on -5x gets us -35x.1851

We subtract by this amount and distribute our negative; it will give us a positive, so 7x3 - 7x3 gets us 0.1859

-43x + 35x gets us -8x; bring down 3x2; bring down 5.1867

We have 3x2 - 8x + 5; at this point, can x3 go into 3x2? No.1876

x3 has a larger degree than 3x2, so we are left with a remainder of 3x2 - 8x + 5.1883

We are left with that remainder; so at this point, we know that our original function, f(x)...1893

through division, we have just shown that it is the same thing as -4x + 7,1897

plus the remainder, 3x2 - 8x + 5, divided by our original denominator, x3 - 5x.1903

So, our answer for what our slant asymptote will end up being is going to be the part in the front, -4x + 7; that is our slant asymptote.1915

Now, we can also check our work, at this point, by just making sure that if we combine these, we get back to our original function.1926

So, let's put them over a common denominator: we have (-4x + 7) times (x3 - 5x), over (x3 - 5x).1933

So, what will that end up being? This is just the same thing here.1948

So, -4x times x3 gets us -4x4; -4x and -5x get us +20x;1951

+7 times x3 gets us 7x3; 7 times -5x gets us -35x; all over x3 - 5x.1962

And we can add on our thing from our other one, because they are now on a common denominator.1974

+ 3x2 - 8x + 5: what does that end up becoming?1978

We are starting to run out of space, so I will do this vertically.1983

We have -4x4 right here; great, that checks out.1986

20...next, x3; 7x3...we have no other x3, so + 7x3; that checks out.1994

3x2...any other x2's?...oops, I accidentally didn't write the squared; -4x times -5x became positive 20x2.2002

So, we have + 3x2 and + 20x2; it becomes + 23x2; that checks out.2010

-35x - 8x...that becomes - 43x; that checks out.2017

And then, + 5...+ 5, and that checks out, because this whole thing is still over that denominator of x3 - 5x.2024

We have the exact same thing that we started with, so what we just did checks out.2035

So, we know for sure that -4x + 7 is good; it is definitely our answer.2041

All right, we will see you next time when we talk about graphing rational functions in general,2045

being able to use these vertical and horizontal asymptotes together to be able to quickly make the graphs to these kinds of functions.2049

All right, see you at Educator.com later--goodbye!2055