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Horizontal Asymptotes (page 2 of 4)

Sections: Vertical asymptotes, Horizontal asymptotes, Slant asymptotes, Examples

Whereas vertical asymptotes are sacred ground, horizontal asymptotes are just useful suggestions. Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes. Whereas vertical asymptotes indicate very specific behavior (on the graph), usually close to the origin, horizontal asymptotes indicate general behavior far off to the sides of the graph. To get the idea of horizontal asymptotes, let's looks at some simple examples.

• Find the horizontal asymptote of the following function:

The horizontal asymptote tells you, roughly, where the graph will go when x is really, really big. So let's look at some big values for x.

x
–100 000	–0.0000099...
–10 000	–0.0000999...
–1 000	–0.0009979...
–100	–0.0097990...
–10	–0.0792079...
–1	0.5
0	2
1	1.5
10	0.1188118...
100	0.0101989...
1 000	0.0010019...
10 000	0.0001000...
100 000	0.0000100...

Do you see how, off to the sides of the graph, where x is strongly negative (such as –1,000) or strongly positive (such as 10000), the "+2" and the "+1" in the expression for y really don't matter so much? Mostly you end up having a really big number divided by a really big number squared, which simplifies to be a very small number. The y-value comes mostly from the "x" and the "x²". And since the x² is "bigger" than the x, the x² drags the whole fraction down to y = 0 (that is, the x-axis) when x gets big.

You can see this behavior on the graph:

As you can see, the graph shows some slightly interesting behavior near the origin, but the rest of the graph is fairly boring, trailing along the x-axis.

In this zoomed-in image, you can also see that the graph crosses the horizontal asymptote (at the arrow):   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

As I mentioned before, it is common and perfectly okay to cross a horizontal asymptote. It's the verticals that you're not allowed to touch.

As we saw from the table of values and from the graph, the horizontal asymptote is the x-axis.

horizontal asymptote:  y = 0 (the x-axis)

In the above problem, the degree on the denominator (2) was bigger than the degree on the numerator (1), and the horizontal asymptote was y = 0 (the x-axis). This property is always true: If the degree on x in the denominator is larger than the degree on x in the numerator, then the denominator, being "stronger", pulls the fraction down to the x-axis when x gets big. That is, if the polynomial in the denominator has a bigger leading exponent than the polynomial in the numerator, then the graph trails along the x-axis at the far right and the far left of the graph.

What happens if the degrees are the same in the numerator and denominator?

• Find the horizontal asymptote of the following:

Unlike the previous example, this function has degree-2 polynomials top and bottom. That is, the degrees are the same in the numerator and the denominator. Since the degrees are the same, the numerator and denominator "pull" evenly; this graph should not drag down to the x-axis, nor should it shoot off to infinity. But where does it go?

Again, think in terms of big values for x. When x is really big, I'll have, roughly, twice something big (minus an eleven) divided by once something big (plus a nine). As you might guess from the last problem, for big values of x, the "–11" and the "+9" won't matter much, so I'll roughly have "2x²/x²", which reduces to just 2. Does a table of values bear this out? Let's check:

x
–100 000	1.9999999...
–10 000	1.9999997...
–1 000	1.9999710...
–100	1.9971026...
–10	1.7339449...
–1	–0.9
0	–1.2222222...
1	–0.9
10	1.7339449...
100	1.9971026...
1 000	1.9999710...
10 000	1.9999997...
100 000	1.9999999...

As you can see, for big values of x, the graph is very close to y = 2.

Then the answer is: horizontal asymptote:  y = 2

In this example, the degree on the numerator and denominator was the same, and the horizontal asymptote was the horizontal line found by dividing the leading coefficients of the two polynomials. This is always true: When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:

y = (numerator's leading coefficient) / (denominator's leading coefficient)

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