HORIZONTAL ASYMPTOTES
In order to determine the horizontal asymptotes of a rational function R(x) =
f (x)
we must evaluate
g(x)
the following limit: lim R(x).
x→∞
2x2 + x − 1
x→∞
x2 + 1
Example: Evaluate lim
Solution: Notice that 2x2 + x − 1 → ∞ and x2 + 1 → ∞ as x → ∞. For this function we can multiply
1
the numerator and denominator by 2 to determine the limit.
x
( )
1
1
1
2
(2x + x − 1)
2+ − 2
2
2x + x − 1
x2
x x
( )
lim
= lim
= lim
1
x→∞
x→∞
x→∞
1
x2 + 1
1+ 2
(x2 + 1)
2
x
x
1
1
− lim 2
2+0−0
x→∞ x
x→∞ x
= x→∞
=
=2
1
1+0
lim 1 + lim 2
x→∞
x→∞ x
1
Here we are exploiting the fact that lim n = 0, where n > 0 is a rational number. Therefore
x→∞ x
2x2 + x − 1
f (x) =
has a horizontal asymptote along the line y = 2 as x approaches ∞.
x2 + 1
lim 2 + lim
x+1
+ 3x − 4
( )
1
1
1
(x + 1)
+ 2
x+1
x2
x x
( ) = lim
lim
= lim
3
4
x→∞ x2 + 3x − 4
x→∞
x→∞
1
1+ 2 − 2
(x2 + 3x + 4)
x
x
x2
Example: Find the horizontal asymptotes of g(x) =
x2
1
1
+ lim
0+0
x→∞ x
x→∞ x
=
=
=0
4
3
1+0−0
lim 1 + lim 2 − lim 2
x→∞ x
x→∞
x→∞ x
lim
We conclude that g(x) has a horizontal asymptote along the line y = 0.
Note: In general for polynomial functions f (x) and g(x) we multiply the numerator and denominator
1
by m where m is the highest power of x that appears in the denominator.
x
Given the rational function R(x) =
f (x)
an xn + an−1 xn−1 + · · · + a1 x + a0
=
,
g(x)
bm xm + bm−1 xm−1 + · · · + b1 x + b0
the horizontal asymptotes for R(x) are determined by the following cases:
(i) If deg(f (x)) > deg(g(x)), R(x) has no horizontal asymptote.
(ii) If deg(f (x)) = deg(g(x)), R(x) has a horizontal asymptote at y =
an
.
bm
(iii) If deg(f (x)) < deg(g(x)), R(x) has a horizontal asymptote at y = 0.
Note: deg(f (x)) and deg(g(x)) represent the degree of f (x) and g(x), respectively.
Examples
(i) R(x) =
x3 + 1
, deg(f (x)) > deg(g(x)). R(x) has no horizontal asymptote.
x2 − 1
(ii) R(x) =
4
y = = 2.
2
(iii) R(x) =
4x2 − 3x + 1
, deg(f (x)) = deg(g(x)). R(x) has an horizontal asymptote along the line
2x2 + 7
x+1
, deg(f (x)) < deg(g(x)). R(x) has an horizontal asymptote along the line y = 0.
x2 + 5x