WEBVTT mathematics/math-analysis/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about horizontal asymptotes.
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In the previous lesson, we learned about the idea of a vertical asymptote,
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a horizontal location where the function blows out to infinity, either up or down.
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Symbolically, we can express this as a vertical asymptote as x approaches some horizontal location, a;
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and when that happens, f(x) goes to positive or negative infinity.
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We can flip this idea to the reverse and discuss the idea of a horizontal asymptote,
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a vertical location which is approached as the horizontal location slides to infinity.
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As our x becomes very, very large, what vertical height do we go to?
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Symbolically, we can express it as x goes to positive or negative infinity (as in, x becomes very, very large,
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either positively or negatively), and f(x) goes to some b, goes to some specific height y = b.
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To understand this, let's take a look at our old friend from last time, our fundamental function, 1/x.
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Notice that, as x grows large, we see f(x) shrink down very small; as we go far out, it becomes very, very small.
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We can see that we are at 1/10 over here and -1/10 over here.
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We can expand this to an even larger viewing window, and we can get a sense for just how small f(x) eventually becomes.
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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.
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We are 1/100 and -1/100, respectively; so we see that becomes really, really, really small, given enough time.
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So, the farther we go out, this f(x) is going to sort of crush down to 0.
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We can see this behavior, being sucked towards a certain height, in many rational functions.
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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.
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Over here, with g(x) = (-3x² + 6)/(x² + 1), it gets pulled towards a height of positive 3.
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In fact, it gets pulled really, really quickly.
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Just like we had with vertical asymptotes, where it never quite touches the asymptote,
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with a horizontal asymptote, it will not quite actually get to there.
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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.
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So, we will formally define this behavior in a little bit; and we will name it a horizontal asymptote.
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But first, let's understand why it occurs: what is actually making this happen?
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Let's start by investigating f(x) = 1/x: since x is in the denominator, as it grows really, really, really large,
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there is a giant denominator that crushes the numerator.
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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.
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So, as it gets really, really big, it crushes the numerator down to 0 in the long run.
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So, if we look at the negative side...over here is negative; we have our vertical asymptote at 0,
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and we are looking at what happens as it slides to the left.
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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.
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At -1000, we have made it to -.001; and it is just going to keep getting smaller and smaller and smaller.
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Negative one billion will be a very, very tiny number.
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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.
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These giant denominators are going to make very, very small numbers that will approach 0.
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We won't actually make it to 0, but we will get really, really, really close to it.
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The same thing happens on the positive side, if we look at what happens as we go positive.
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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.
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At 5, we are at 0.2, and so on; we get that, at 1000, we are at .001.
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And we as we get really, really large numbers, we will get crushed smaller and smaller and smaller.
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Since the denominator grows so much faster than the numerator (the numerator isn't moving at all,
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so it is not even growing at all), the fraction will eventually shrink to 0, as we get very large denominators crushing our numerator.
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For any rational function, if the denominator's degree is greater than the numerator's degree--
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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.
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If we have x²/(x³ + 5), that is eventually going to get crushed down to 0,
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because x² doesn't grow as fast as x³ + 5.
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So, the x³ + 5 can effectively outrun x² in the long run, so it will get much larger than x² will.
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And so, it will crush the whole thing down to 0.
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So, if the degree is greater in the denominator (3 versus the numerator's degree, 2), it will eventually get crushed to 0.
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How do we get rational functions with a horizontal asymptote that isn't at 0, then?
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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.
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If we plug in 1 (once again, this will be the negative side), what happens as we go more and more negative?
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If we plug in +1, we will get 5(1)/(1 - 2), so we get 5/-1, which is -5.
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If we plug in -10, we will get -50/-12, approximately 4.17.
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-100 gives -500/-102, which is approximately 4.90; -1000 gives -5000/-1002, which is 4.99, approximately.
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So, notice: as we plug in these things, the 5x here and the positive x here,
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they end up growing at the same rate, other than this multiplicative factor of 5.
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So, the top grows 5 times faster, precisely 5 times faster, than the bottom does.
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So, as they go out one way or the other, they are going to end up approaching...the top is growing
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5 times faster than the bottom is growing, so it is going to end up approaching 5,
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because when we have very, very large numbers that we are going to put in (eventually, like 1 million),
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it will be 5(really big number), divided by (really big number); so it will cancel out to 5.
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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.
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It has a slight effect, but it is not much of an effect.
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And so, as we get down to farther and farther values out, as we get to larger and larger values,
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it will have less and less of a relative effect, and we are going to get closer and closer to 5.
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The exact same thing happens if we look at what happens on the positive side.
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If we start by plugging in 3, we are at 15/1, so we are very different at 3; we are at 15.
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But when we plug in 10, we are at 50/8; at 100, 500/98; at 1000, 5000/998.
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We have this difference that becomes less and less impressive; this -2 becomes less and less meaningful.
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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.
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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.
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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.
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So, it won't ever actually equal that horizontal asymptote precisely.
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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.
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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.
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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.
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So, for any rational function, if the degrees of the numerator and the denominator are equal,
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we will get a horizontal asymptote that isn't equal to 0; we will get a horizontal asymptote at some height.
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Notice: the 5x here and the x here have the same degree, a degree of 1.
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So, since they have the same degree on the top and the bottom, they are going at the same rate, in a way.
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Other than that multiplicative factor of 5, they are running in the same scale of growth.
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So, since they have the same scale of growth, they are going to grow around the same rate,
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which means it is only that multiplicative factor that is going to determine the height that it ends up at.
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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.
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Symbolically, we show this as "x goes to negative infinity" or "x goes to positive infinity"
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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.
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It might not get perfectly to b; in fact, it almost certainly won't, as what we were just talking about.
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But it will get really, really close to b; it will get arbitrarily close to it.
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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.
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Over the long term, it will start somewhere else, but it gets pulled, in the long term, to a certain height,
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until it gets really, really close to this horizontal asymptote.
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We can take this idea and go beyond just having a horizontal asymptote.
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Consider f(x) = (x³ - 1)/x²; as x gets large, we see f(x) grow very close to the line y = x.
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We can see that on that dashed orange line going through the y = x line.
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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.
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It will never be perfectly the same, because we have this -1 here; but it will become really, really close to it.
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This idea is similar to a horizontal asymptote, but it is no longer horizontal.
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Since it is at a slant, we can't call it a horizontal asymptote; so instead, we call it a **slant asymptote**.
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Sometimes, it is also called an **oblique asymptote**.
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So, what is going on--why do we see this?
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Once again, we are trying to consider what happens to the function in the long run.
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The idea of all this horizontal asymptote/slant asymptote stuff is a question of what happens to this function
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as we look at very large x--as we go really far right/as we go really far left.
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In this case, we could plug in large numbers to see what happens; but that will slightly obscure some details for other slant asymptotes.
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So, instead, what we want to notice is that we can rewrite the function.
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If we have (x³ - 1)/x², we could say, "Look, we can divide out the x³,
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and we can break our fraction apart so that we get x³/x² - 1/x²."
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And so, the x³ and x² cancel down to just x - 1/x².
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So, by using division, we can see this function in a new way.
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In this form, it is clear that, as x goes to positive or negative infinity, this part right here,
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since it is 1/x², is just going to sort of get crushed down to 0 by its much larger denominator.
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But the x here will end up just continuing on; it will just keep moving forward.
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So, in the long run, as we get to very large x's, this part here goes to 0; but x continues going out.
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So, effectively, the function will become just x.
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We can also get this (x³ - 1)/x²...we can break it into this format
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through the process of polynomial long division, which will be necessary
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when we have slightly more complicated denominators that we are dealing with.
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So, you write this as this, minus 1; so in that way, we would have x².
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How many times does x² go into x³?
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It goes in just x; x times x² gets us x³.
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We have nothing else, so we subtract by x³; that gets us 0.
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We bring everything down; and so we have 0x² + 0x - 1, which leaves us with just a remainder of -1.
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We have this remainder of -1; so (x³ - 1)/x² is equal to x, plus the remainder.
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The remainder is -1; and then we put it back over our original denominator, the thing that we divided by.
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We get x plus -1 over x², which is the exact same thing that we had down here on this line.
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We can get this through polynomial division, which will be necessary when we are dealing with slightly more complicated denominators.
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We can also define a slant asymptote similarly to how we defined a horizontal one.
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It is also called an oblique asymptote sometimes.
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A slant asymptote is a line y = mx + b (remember, mx + b is just our normal slope-intercept form for a line),
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where, as x becomes very large (positive or negative), f(x) gets arbitrarily close to that line, y = mx + b.
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Symbolically, we can show this as: as x approaches negative infinity (very large negative values),
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or x approaches positive infinity (very large positive values), f(x) will go to mx + b.
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f(x) will approach just being the same as this line, mx + b.
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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.
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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.
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We can even go beyond the idea of a slant asymptote.
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Really, the question we have been working on can be phrased as "What does the function look like in the long term?"
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What is the long-term behavior of this function?
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So far, we have answered that with horizontal and slant asymptotes.
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But a function could also tend to a curve; it could tend to anything, really.
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If we had (x⁴ + 17x + 20)/(10x² - 10x - 20), well, notice:
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we could make that as squared and to the fourth up here.
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So, the degree of the top is one, two steps larger than the degree on the bottom.
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That means that, over time, it is going to effectively be the same as x² coming out of that, x⁴ divided by x².
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We are going to get something that looks kind of like an x², which is a parabola, which is exactly what we see here.
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As we get to very large x-values, we see it get pulled along this curve.
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Notice this curve that we have of a parabola through here.
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Now, it will behave differently when we are at the vertical asymptotes, because we have vertical asymptotes x - 2 and x + 1.
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So, there are vertical asymptotes at -1 and vertical asymptotes at +2.
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We will get pulled into these vertical asymptotes in various ways.
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But in the long run, as it gets to very large x-values, it gets pulled into this parabolic shape.
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In fact, if we were to divide this out through dividing the bottom into the top through polynomial division,
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we would be able to find that it eventually is approaching the parabola 1/10 times x² plus x plus 3.
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And that is why we see that parabolic curve right there--pretty cool.
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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.
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While this shows us an interesting idea, don't expect to see this in a normal class.
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Few textbooks and very few teachers will discuss anything beyond the idea of a slant asymptote.
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As such, we will not be exploring the idea any further in this class, either.
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However, it is useful to notice how all of these ideas have been linked.
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They are about answering: "Where does this go in the long term? What is happening eventually to my function?
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How will it behave when I look at very large values being plugged in?"
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We can get a sense of this by thinking about what happens to the function as the numbers get larger and larger and larger.
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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?
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That is what all of these ideas in this lesson have been about.
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What happens as x becomes very, very large--how does this thing behave?
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It could behave in these non-slant asymptote things where it pulls into a parabola or some other polynomial shape.
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But we are going to restrict ourselves to just horizontal and slant asymptotes, since that is what most other courses look at.
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And they are also the easiest for us to approach right now.
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Horizontal/slant asymptotes and graphs: Just like their vertical cousins, it is customary to show horizontal/slant asymptotes with a dashed line.
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We normally use a dashed line to show, "Look, here is an asymptote."
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So, if we had (4x⁴ + 3x³ + 10x²)/(2x⁴ + 1), we would get this graph over here on the right.
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And notice: it has a horizontal asymptote at 2, and so, over the long run, our graph gets pulled towards this.
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Now, notice that, in the middle, it has a behavior that is totally different.
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It has this interesting behavior in the middle.
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Unlike vertical asymptotes, the graph can cross the horizontal asymptote.
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It is allowed to actually cross over that horizontal or slant asymptote.
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Furthermore, there is only ever one horizontal or slant asymptote.
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You can't have multiple horizontal/slant asymptotes, in the same way you can't have a vertical asymptote.
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In any case, over the long run, the graph will be pulled along the asymptote.
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That is the idea of an asymptote to really get across: that an asymptote is about the function eventually being pulled along it,
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or, if it is a vertical asymptote, being stretched up along it vertically.
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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.
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It is fairly easy to determine if a function has a horizontal asymptote, and, if so, what it is.
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We will see a method for that first.
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Finding a slant asymptote is a little bit trickier, though, and we will look at its method second; but it is not that difficult.
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Any rational function is in the form n(x)/d(x), where they are both polynomials.
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We can expand these polynomials into their normal form, our _x^n + _x^n - 1 + _x^n - 2,
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all the way until we eventually hit a constant; and the bottom one will be the same thing--some other blank.
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So, a<font size="-6">n</font> will be the blank on the top; b<font size="-6">n</font> will be the blank on the bottom.
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And we have two different things; n will be the numerator's degree, and m will illustrate the denominator's degree.
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There are going to be three possibilities: first, if n is less than m, then there is a horizontal asymptote at y = 0.
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Why is that the case? Well, if n is less than m, then that means we have a big denominator, compared to the numerator.
00:18:03.400 --> 00:18:07.300
The numerator will have a smaller degree; so it is going to grow slower,
00:18:07.300 --> 00:18:11.600
which means that the bottom one will eventually grow large enough to crush the numerator.
00:18:11.600 --> 00:18:14.200
So, we are going to crush it down to y = 0.
00:18:14.200 --> 00:18:20.200
So, if we have a numerator degree that is less than the degree of the denominator,
00:18:20.200 --> 00:18:25.600
the denominator will eventually grow large enough to crush the whole fraction down to 0.
00:18:25.600 --> 00:18:34.300
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:
00:18:34.300 --> 00:18:38.100
we will see a horizontal asymptote that is based on the ratio of the leading coefficients.
00:18:38.100 --> 00:18:41.100
We will get a horizontal asymptote, but it will no longer be set at y = 0.
00:18:41.100 --> 00:18:44.900
Since x^n and x^m are basically the things that, in the long run,
00:18:44.900 --> 00:18:49.500
are really going to determine how these polynomials are, and n = m,
00:18:49.500 --> 00:18:53.600
then ultimately it is going to be a<font size="-6">n</font>x^n/b<font size="-6">m</font>x^m,
00:18:53.600 --> 00:18:57.000
in the long run, since x^n and x^m are the same value.
00:18:57.000 --> 00:19:01.700
Since those things are at the front, they are going to really determine how the polynomial works in the long run.
00:19:01.700 --> 00:19:06.700
And they will just cancel each other out, because we will have a<font size="-6">n</font>x^n/b...
00:19:06.700 --> 00:19:11.500
and I will call it n as well, since we have n equal to m...x^n.
00:19:11.500 --> 00:19:15.300
Well, in the long run, what is effectively going to happen is that we will see these two things cancel each other.
00:19:15.300 --> 00:19:21.300
And we will be left with just the ratio of the leading coefficients.
00:19:21.300 --> 00:19:25.900
That will determine what the horizontal asymptote is going to go to--this ratio.
00:19:25.900 --> 00:19:29.800
What is the leading coefficient on the top, divided by the leading coefficient on the bottom?
00:19:29.800 --> 00:19:33.500
What is our first coefficient here and our first coefficient there?
00:19:33.500 --> 00:19:45.600
Finally, if n is greater than m, if we have the numerator degree bigger, then there is no horizontal asymptote,
00:19:45.600 --> 00:19:49.000
because the numerator is able to run faster than the denominator.
00:19:49.000 --> 00:19:55.300
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.
00:19:55.300 --> 00:20:00.200
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,
00:20:00.200 --> 00:20:04.900
as it manages to have very large values, because it will be able to outrun the denominator, because it has a larger degree.
00:20:04.900 --> 00:20:08.000
All right, so let's talk about how to find slant asymptotes.
00:20:08.000 --> 00:20:14.900
A rational function has a slant asymptote if the degree of the numerator is exactly 1 greater than the degree of the denominator.
00:20:14.900 --> 00:20:22.600
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.
00:20:22.600 --> 00:20:24.900
We can find the asymptote through polynomial division.
00:20:24.900 --> 00:20:30.100
For example, if we have (3x³ - 2x² + 7x + 8)/(x² - 3x), we say,
00:20:30.100 --> 00:20:40.000
"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."
00:20:40.000 --> 00:20:47.600
So, we are exactly 1 greater in the degree of the numerator than we are in the denominator.
00:20:47.600 --> 00:20:49.700
So, we are going to have a slant asymptote.
00:20:49.700 --> 00:20:54.400
At that point, we use polynomial division; so let's see how polynomial division would work here.
00:20:54.400 --> 00:21:08.300
We have x² - 3x as dividing into...what is in our numerator? 3x³ - 2x² + 7x + 8.
00:21:08.300 --> 00:21:10.700
So, how many times does x² go into 3x³?
00:21:10.700 --> 00:21:16.900
It is going to go in 3x; 3x times x² gets us 3x³, so yes, we were right.
00:21:16.900 --> 00:21:24.300
3x times -3x gets us -9x²; then we subtract this whole thing, so let's distribute that negative:
00:21:24.300 --> 00:21:30.900
minus 3x²...that becomes + 9x²; so 3x³ - 3x³ is 0.
00:21:30.900 --> 00:21:35.100
-2x² + 9x² is positive 7x².
00:21:35.100 --> 00:21:44.900
The next step: bring down the 7x, so + 7x; how many times does x² go into 7x²? It goes in + 7 times.
00:21:44.900 --> 00:21:50.800
So, 7x²...that checks out; 7 times -3 is -21x.
00:21:50.800 --> 00:22:01.000
We subtract that whole thing and distribute our negative: minus, plus...7x² - 7x² is 0; 7x + 21x is 28x.
00:22:01.000 --> 00:22:09.800
Bring down the 8; we get + 8 here; at this point, we see 28x--how many times can x² go into 28x?
00:22:09.800 --> 00:22:17.200
It can't go in anymore because of our degrees; so we have a remainder of 28x + 8.
00:22:17.200 --> 00:22:23.800
And so, notice how these are the same thing: 3x + 7 is what we got as the result, 3x + 7 here.
00:22:23.800 --> 00:22:33.600
And then, plus our remainder, 28x + 8, divided by what we started doing our division with...
00:22:33.600 --> 00:22:38.400
that is how we get polynomial division; and indeed, 3x + 7 + (28x + 8)
00:22:38.400 --> 00:22:48.500
divided by (x² - 3x) is what our initial rational function is equal to; so we can break it down.
00:22:48.500 --> 00:22:56.800
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).
00:22:56.800 --> 00:23:08.200
So, that is what our slant asymptote is, because 3x + 7 describes a line that is in the form mx + b.
00:23:08.200 --> 00:23:23.800
So, 3x + 7 is a line, and the remainder to our division is our 28x + 8; it goes over the denominator x² - 3x,
00:23:23.800 --> 00:23:30.600
back over the denominator that we started with; and notice, (28x + 8)/(x² - 3x), because it is the remainder...
00:23:30.600 --> 00:23:34.300
the remainder is always going to have a degree less than what we started dividing with.
00:23:34.300 --> 00:23:40.900
So, we have 28x¹ divided by (x² - 3x); so we have a bigger degree on the bottom than we do on the top.
00:23:40.900 --> 00:23:49.600
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.
00:23:49.600 --> 00:23:55.100
So, in the long run, we will end up having a portion that has no denominator, and the portion that has a denominator,
00:23:55.100 --> 00:24:01.500
because the degree on the bottom is now less after polynomial division--it will go to 0 in the long term.
00:24:01.500 --> 00:24:06.100
Now, this method of polynomial division will also work to find horizontal asymptotes.
00:24:06.100 --> 00:24:09.300
So, you can also use this method if you want to find the horizontal asymptote.
00:24:09.300 --> 00:24:18.400
It is just that, instead of getting a line here, you will just get a constant--it will just be a constant value,
00:24:18.400 --> 00:24:22.700
if it is a horizontal asymptote, as opposed to a slant asymptote.
00:24:22.700 --> 00:24:26.100
So, you can also use polynomial division to find horizontal asymptotes.
00:24:26.100 --> 00:24:29.200
But we have that other method for finding horizontal asymptotes that was pretty fast.
00:24:29.200 --> 00:24:33.700
So, it is normally easiest to just use polynomial division when you want slant asymptotes.
00:24:33.700 --> 00:24:35.500
All right, let's go over some examples.
00:24:35.500 --> 00:24:43.200
f(x) = (10x⁵ + 3x⁴ + 8x² - 2)/(2x⁵ + 27x³ + 12x).
00:24:43.200 --> 00:24:45.400
Is there a horizontal or a slant asymptote?
00:24:45.400 --> 00:24:48.900
What we do is compare the degree on the top to the degree on the bottom.
00:24:48.900 --> 00:24:56.700
They are the same degree; so if they are the same degree, we have a horizontal asymptote.
00:24:56.700 --> 00:25:01.700
Now, if we want to figure out what the horizontal asymptote is, well, we will figure that out.
00:25:01.700 --> 00:25:11.500
And we just look at its ratio of leading coefficients; what is the leading coefficient for the top and the bottom?
00:25:11.500 --> 00:25:20.700
The top has a leading coefficient of 10; the bottom has a leading coefficient of 2; we simplify that, and we get 5.
00:25:20.700 --> 00:25:30.800
So, 5 is what our horizontal asymptote will be; so y = 5 is the horizontal asymptote.
00:25:30.800 --> 00:25:38.700
Great; once we see that the degree on the top and the degree on the bottom are the same,
00:25:38.700 --> 00:25:42.000
we know we have a horizontal asymptote that is not just going to be a 0.
00:25:42.000 --> 00:25:45.100
And now, we figure it out by looking at the ratio of the leading coefficients,
00:25:45.100 --> 00:25:49.900
because ultimately, the ratio of the leading coefficients on our biggest exponent, x's,
00:25:49.900 --> 00:25:54.500
is going to be what determines what happens to these functions in the long run.
00:25:54.500 --> 00:25:58.000
The next one: Using the graph of the function, determine a.
00:25:58.000 --> 00:26:04.400
We have (12x³ + 5x² - 10x + 8)/(ax³ + 2x - 2).
00:26:04.400 --> 00:26:11.300
Now, we notice that there are 3 and 3; and here is a horizontal asymptote.
00:26:11.300 --> 00:26:15.300
We have a horizontal asymptote, because the degrees are the same.
00:26:15.300 --> 00:26:18.700
Also, we can see that we have a horizontal asymptote in our graph,
00:26:18.700 --> 00:26:22.600
so it had better be the fact that the degree on the numerator and the denominator is the same.
00:26:22.600 --> 00:26:26.700
At this point, we know...what is our horizontal asymptote? It is y = 3.
00:26:26.700 --> 00:26:32.200
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.
00:26:32.200 --> 00:26:38.300
So, if that is the case, we know that the ratio of the leading coefficients, 12/a
00:26:38.300 --> 00:26:46.700
(the leading coefficient on top is 12; on the bottom, it is a), must be equal to 3, our horizontal asymptote.
00:26:46.700 --> 00:26:56.100
We work this out: 12 = 3a; divide by 3, and we get 4 = a, so a is 4; great.
00:26:56.100 --> 00:27:04.300
We have (28x³ + 110x² - 47x + 55)/(0.2x⁴ - 5x² + 4).
00:27:04.300 --> 00:27:07.600
Is there a horizontal or a slant asymptote? What is it?
00:27:07.600 --> 00:27:10.900
In this case, the first thing, as always: we look to see what are our degrees.
00:27:10.900 --> 00:27:14.800
The numerator degree is 3; the denominator degree is a 4.
00:27:14.800 --> 00:27:18.900
So, if the denominator degree is larger, it crushes everything.
00:27:18.900 --> 00:27:22.700
It is going to be able to eventually, in the long run, overtake the numerator.
00:27:22.700 --> 00:27:27.100
It will run faster than the numerator and grow larger, and it will crush everything to 0.
00:27:27.100 --> 00:27:32.600
So, there is a horizontal asymptote, but it is going to be the most boring horizontal asymptote of all,
00:27:32.600 --> 00:27:37.300
but at the same time, kind of interesting: y = 0.
00:27:37.300 --> 00:27:43.400
The denominator crushes that puny numerator; that numerator is just too small,
00:27:43.400 --> 00:27:47.800
so the denominator crushes the puny numerator, because it has a larger degree.
00:27:47.800 --> 00:27:54.200
And I want to point out that it seems at first...well, we have .2x⁴ and 28 times x³;
00:27:54.200 --> 00:27:58.100
and there are also...110 and these other big numbers up here on the top.
00:27:58.100 --> 00:28:01.100
But on the bottom, it all seems like pretty small, insignificant numbers.
00:28:01.100 --> 00:28:06.200
So, why is it that the bottom is going to be bigger?...because it has a larger degree.
00:28:06.200 --> 00:28:11.400
Ultimately, how a polynomial behaves in the long run is really determined by that degree.
00:28:11.400 --> 00:28:19.000
The coefficients have effects; they affect things; but being really dominant is just determined by having the biggest degree.
00:28:19.000 --> 00:28:22.500
x⁴, no matter how small that coefficient at the front is,
00:28:22.500 --> 00:28:27.100
will eventually be able to outrun x³, no matter how big its coefficient is at the front.
00:28:27.100 --> 00:28:32.600
So, that x⁴ will crush the numerator, because the numerator only has a degree of 3.
00:28:32.600 --> 00:28:40.100
All right, the final example: at this point, we have (-4x⁴ + 7x³ + 23x² - 43x + 5)/(x³ - 5x).
00:28:40.100 --> 00:28:43.400
We are asked, "Is there a horizontal or slant asymptote?" and then, "What is it?"
00:28:43.400 --> 00:28:48.400
The first thing we do is look at our degrees: we have 4 on the numerator and 3 on the denominator.
00:28:48.400 --> 00:29:01.100
So, that means that the degree of the numerator is exactly 1 greater than the denominator.
00:29:01.100 --> 00:29:07.100
If it is 1 greater, then that means we have a slant asymptote; and that makes sense,
00:29:07.100 --> 00:29:13.000
because one degree larger than something else...if we divide them out...x^50/x^49...
00:29:13.000 --> 00:29:15.600
that is going to become something along the lines of x.
00:29:15.600 --> 00:29:19.500
So, it is going to have a nice linear form; it is going to become a line.
00:29:19.500 --> 00:29:25.300
Degree 1 is a linear function, so that is why we see a slant asymptote, a line, coming out of it.
00:29:25.300 --> 00:29:29.100
How do we figure it out? We figure it out by using polynomial division.
00:29:29.100 --> 00:29:35.600
So, (x³ - 5x)...and notice: that is 5x¹, not 5x²;
00:29:35.600 --> 00:29:39.300
that is going to have a slight effect when we are doing our division.
00:29:39.300 --> 00:29:50.700
-4x⁴ + 7x³ + 23x² - 43x + 5.
00:29:50.700 --> 00:29:53.300
x³: how many times does it go into -4x⁴?
00:29:53.300 --> 00:30:00.600
It is going to go in -4x times, because -4x times x³ gets us -4x⁴.
00:30:00.600 --> 00:30:08.300
-4x times -5x gets us 20x²; but the next thing we have is 7x³, so that is not going to go on the 7x³.
00:30:08.300 --> 00:30:12.000
It is going to go on the 23x² column.
00:30:12.000 --> 00:30:17.800
-4x times -5x gets us 20x²; so 20x² lines up there.
00:30:17.800 --> 00:30:21.100
And it is positive; so at this point, we subtract by this amount.
00:30:21.100 --> 00:30:25.000
So, we distribute our negative; that becomes positive; that becomes negative.
00:30:25.000 --> 00:30:31.300
-4x⁴ + 4x⁴ becomes 0 (what it should be when we are doing polynomial division--the first part should always cancel to 0).
00:30:31.300 --> 00:30:36.500
23x² - 20x² gets us positive 3x².
00:30:36.500 --> 00:30:41.600
The next step: we bring down the other things that we will end up using.
00:30:41.600 --> 00:30:47.700
We bring down the 7x³; we bring down the -43x.
00:30:47.700 --> 00:30:51.500
How many times does x³ go into 7x³? It will go in 7 times.
00:30:51.500 --> 00:30:59.300
7 times x³ gets us 7x³; 7 on -5x gets us -35x.
00:30:59.300 --> 00:31:07.400
We subtract by this amount and distribute our negative; it will give us a positive, so 7x³ - 7x³ gets us 0.
00:31:07.400 --> 00:31:16.500
-43x + 35x gets us -8x; bring down 3x²; bring down 5.
00:31:16.500 --> 00:31:23.300
We have 3x² - 8x + 5; at this point, can x³ go into 3x²? No.
00:31:23.300 --> 00:31:33.000
x³ has a larger degree than 3x², so we are left with a remainder of 3x² - 8x + 5.
00:31:33.000 --> 00:31:37.400
We are left with that remainder; so at this point, we know that our original function, f(x)...
00:31:37.400 --> 00:31:42.900
through division, we have just shown that it is the same thing as -4x + 7,
00:31:42.900 --> 00:31:55.400
plus the remainder, 3x² - 8x + 5, divided by our original denominator, x³ - 5x.
00:31:55.400 --> 00:32:06.300
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.
00:32:06.300 --> 00:32:12.800
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.
00:32:12.800 --> 00:32:27.700
So, let's put them over a common denominator: we have (-4x + 7) times (x³ - 5x), over (x³ - 5x).
00:32:27.700 --> 00:32:30.800
So, what will that end up being? This is just the same thing here.
00:32:30.800 --> 00:32:42.100
So, -4x times x³ gets us -4x⁴; -4x and -5x get us +20x;
00:32:42.100 --> 00:32:53.800
+7 times x³ gets us 7x³; 7 times -5x gets us -35x; all over x³ - 5x.
00:32:53.800 --> 00:32:57.900
And we can add on our thing from our other one, because they are now on a common denominator.
00:32:57.900 --> 00:33:03.300
+ 3x² - 8x + 5: what does that end up becoming?
00:33:03.300 --> 00:33:05.700
We are starting to run out of space, so I will do this vertically.
00:33:05.700 --> 00:33:14.000
We have -4x⁴ right here; great, that checks out.
00:33:14.000 --> 00:33:22.200
20...next, x³; 7x³...we have no other x³, so + 7x³; that checks out.
00:33:22.200 --> 00:33:29.700
3x²...any other x²'s?...oops, I accidentally didn't write the squared; -4x times -5x became positive 20x².
00:33:29.700 --> 00:33:37.100
So, we have + 3x² and + 20x²; it becomes + 23x²; that checks out.
00:33:37.100 --> 00:33:43.700
-35x - 8x...that becomes - 43x; that checks out.
00:33:43.700 --> 00:33:55.300
And then, + 5...+ 5, and that checks out, because this whole thing is still over that denominator of x³ - 5x.
00:33:55.300 --> 00:34:00.800
We have the exact same thing that we started with, so what we just did checks out.
00:34:00.800 --> 00:34:04.700
So, we know for sure that -4x + 7 is good; it is definitely our answer.
00:34:04.700 --> 00:34:08.900
All right, we will see you next time when we talk about graphing rational functions in general,
00:34:08.900 --> 00:34:14.900
being able to use these vertical and horizontal asymptotes together to be able to quickly make the graphs to these kinds of functions.
00:34:14.900 --> 00:34:16.000
All right, see you at Educator.com later--goodbye!