Lesson Two - Horizontal Asymptotes
Complete the table of values for f ( x ) =
x
0
1
2
5
 10
 100
 1000
x2 + 2
.
x2 + 1
y
What do we see? As x → , y → 1 . Therefore the horizontal asymptote is y = 1.
Horizontal Asymptotes
➢ Looking at what is happening on the graph when x is approaching very
large and very small values ( ±∞)
➢ Not dealing with when the function is undefined, therefore there could
be times when the graph WILL cross a horizontal asymptote
➢ To determine horizontal asymptotes we need to look at the DEGREE of
the function in the numerator AND the denominator
When graphing functions with horizontal asymptotes NEED TO:
1) SUB IN a large and small value of x (100, -100) to see if the function is above or
below your asymptote
2) CHECK if the function crosses horizontal asymptote by making the value of the
horizontal asymptote equal to the function (Point of intersection). If you can
solve for x, the function crosses at that point.
NOTE: A graph may have both a vertical and horizontal asymptote.
1. Find the horizontal and vertical asymptotes. Sketch the curve near the asymptotes.
Check to see if the graph crosses the asymptote at any specific point.
a) f ( x ) =
1
x2
x +1
f
(
x
)
=
b)
x−2
4 x 2 − 3 x + 21
c) f ( x) =
2 x2 + 5