Section 4.2

More on Graphs of Rational Functions
HORIZONTAL ASYMPTOTES:
Note the graph of
1
( )
f x
x
=
As x gets very large in value, x
+∞
(approaches positive infinity), the graph gets
closer to (approaches) the line
y = 0 (the xaxis). It approaches the line y = 0
asymptotically. (An asymptote is a line a curve approaches infinitely along its length.).
The graph also approaches the line y = 0 as x
∞
.
The line y=0 is called the
horizontal asymptote
. And since the graph also vertically
approaches the line x=0, it is called the
vertical asymptote
.
We will be graphing rational functions by noting their horizontal and vertical asymptotes.
DEFINITION:
A rational function
( )
( )
( )
g x
f x
q x
=
has a
horizontal asymptote
at
y = b if
the values of f(x) approach the value
b
as
x
becomes increasingly large.
Ex.
6
( )
2
x
f x
x
=

Note: x can not = 2!
Try some values > 2.
x
3
4
5
6
8
10
100
1000
f(x)
18
12
10
9
8
7.5
6.1+
6.012+
So as x gets very large, the function approaches 6 (from slightly
above
6).
Try some values < 2.
x
1
0
4
8
10
100
1000
f(x)
6
0
4
4.8
5
5.88+
5.988+
So as x gets very large negatively, the function approaches 6 (from slightly
below
6).
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View Full DocumentSo from the above definition, the line
y = 6 is a horizontal asymptote for this function.
The function will approach the line
y = 6 on the “
extreme ends
” of the graph. See the
graph below.
Graph by itself.
Graph with H.A. : y = 6
We have something called a “limit” statement to indicate this result. The graph
approaches the value 6 as a limit on each end of the graph. The statements look like :
lim
( )
6
x
f x
→+∞
=
and
lim
( )
6
x
f x
→∞
=
.
This says “The limit of the function as x approaches
positive infinity is 6.”
Also, “The limit of the function as x approaches negative infinity
is 6.”
Again, this means the graph will approach the horizontal value y=6 on each end of
the graph.
VERTICAL ASYMPTOTES:
A
vertical asymptote
is a vertical line ( x = ?) that the graph will approach (but not
touch!). A vertical asymptote will occur at a value of x that makes ONLY the
denominator of the function = 0.
Ex.
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 Fall '06
 Lebre
 Algebra, Asymptotes, Rational Functions, Fraction, Limit of a function, Rational function

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