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 me, roughly, where the graph will go when x
is really, really big. So I'll look at some very big values for x,
some values of x
very far from the origin:

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...

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. I ended up having a really big number divided
by a really big number squared, which "simplified" to be a
very small number. The y-value
came mostly from the "x"
and the "x^{2}".
And since the x^{2}
was "bigger" than the x,
the x^{2}
dragged the whole fraction down to
y = 0 (that
is, the x-axis)
when x
got big.

I can
see this behavior on the graph:

The graph shows some
slightly interesting behavior in the middle, near the origin, but the
rest of the graph is fairly boring, trailing along the x-axis.

(It is common and perfectly
okay to cross a horizontal asymptote. It's the vertical asymptotes that
I'm not allowed to touch.)

As I can see in the table
of values and the graph, the horizontal asymptote is the x-axis.

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

In the above exercise,
the degree on the denominator (namely, 2)
was bigger than the degree on the numerator (namely, 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:

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Unlike the previous example,
this function has degree-2
polynomials top and bottom; in particular, 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 will it go?

Again, I need to 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 exercise, the "–11"
and the "+9"
won't matter much for really big values of x.
Far off to the sides of the graph, I'll roughly have "2x^{2}/x^{2}",
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...

For big values of x,
the graph is, as expected, very close to y
= 2.

The graph reflects
this:

Then my answer is: horizontal
asymptote: y
= 2

In the example above, the
degrees on the numerator and denominator were the same, and the horizontal
asymptote turned out to be the horizontal line whose y-value
was equal to the value 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)