Limits at Infinity for Rational Functions
The sort of rational functions we will often deal with in this course are those
which are ratios of polynomials, written as
f (x) =
P(x)
a x m + bx m−1 + K + k
=
Q(x)
px n + qx n−1 + K + z
,
where the coefficients a , b , … , k , p , q , … , z are real constants with a ≠ 0 and p ≠ 0
and m and n are positive integers. Here, we shall be concerned with the “limit at
infinity” of such
€ functions.
It will be simplest to investigate first the limit at positive infinity of these
functions. We cannot simply apply the regular “Limit Laws” because that leads to the
indeterminate ratio
a x m + bx m−1 + K + k
∞+ ∞+K+ k
∞
lim f (x) = lim
= "
" = " " .
n
n−1
∞+ ∞+K+ z
∞
+K+ z
x →∞
x →∞ px + qx
€
To avoid this situation, we would like to arrange for the denominator to produce a finite
number which is not zero. We can accomplish this by dividing the numerator and
denominator of this ratio by the largest power of x appearing in the denominator, xn ,
1
and then applying the special “Limit Law”, lim n = 0 , for n > 0 , to the resulting
x →∞ x
expression. We will need to consider three cases:
€
Case I : the polynomials P(x) and Q(x) have equal degree ( m = n ) –
m
1
m−1
a
xm
+b
a x + bx
+ K + k xm
xm
lim
⋅
=
lim
m
m−1
m
+ K + z 1m
x →∞ px + qx
x →∞ p x
+q
x
xm
a + b lim
€
=
x →∞
p + q lim
x →∞
€
1
x1
1
x1
+ c lim
x →∞
+ r lim
x →∞
1
x2
+ K + k lim
1
+K+
x2
k
+K+ m
x
m−1
x
z
m +K+ m
xm
x
x
1
m
x →∞ x
1
z lim
x m−1
m
x →∞ x
=
a + b ⋅ 0 + c ⋅ 0 +K+ k ⋅ 0
a
=
.
p + q ⋅ 0 + r ⋅ 0 +K+ z ⋅ 0
p
Case II : the polynomial Q(x) have the higher degree ( m < n or m – n < 0 ) –
xm
x m−1
k
+K+ n
n
a x + bx
+K+ k
x
x
lim
⋅ 1 = lim
n
n−1
n
n−1
x
x
z
px
+
qx
+
K
+
z
x →∞
x →∞ p
+ q n +K+ n
xn
n
m
1
m−1
a
xn
+b
xn
x
€
1
p +q⋅
x →∞
x1
1
a lim
n−m
x →∞ x
=
€
p + q lim
1
x2
+K+
1
x1
1
+K+
x2
x →∞
xn
1
+ r lim
xn
z
+ b lim
n−m +1
x →∞ x
x →∞
+r⋅
x
k
a ⋅ x m−n + b ⋅ x m−n−1 + K +
= lim
x
1
+ K + k lim
n
x →∞ x
1
z lim
xm
x →∞
1 
 x m−n = x −(n−m ) = n−m

x
with n – m > 0
€
=
a ⋅ 0 + b ⋅ 0 + c ⋅ 0 +K+ k ⋅ 0
0
=
= 0 .
p + q ⋅ 0 + r ⋅ 0 +K+ z ⋅ 0
p€
Case III : the polynomial P(x) have the higher degree ( m > n or m – n > 0 ) –
€
since m > n , there is some term
with exponent n
m
1
m−1
a x + bx
+ K + k xn
lim
⋅
= lim
n
n−1
+ K + z 1n
x →∞ px + qx
x →∞
x
€
= lim
a
xm
xn
+b
p
x m−1
xn
xn
xn
+q
a ⋅ x m−n + b ⋅ x m−n−1 + K + C ⋅ x 0 + K +
x →∞
p +q⋅
1
x1
+r⋅
1
x2
+K+
z
+K+ C
x n−1
=
x →∞
p + q lim
x →∞
x →∞
1
+
x1
r lim
x →∞
1
+K+
x2
xn
+K+
+K+
k
xn
z
xn
k
xn
xn
a lim x m−n + b lim x m−n−1 + K + C + K + k lim
€
xn
xn
z lim
x →∞
x →∞
1
1
xn
xm
€
€
=
a ⋅∞+ b ⋅∞+K+ C +K+ k ⋅ 0
a ⋅∞
= "
" = ∞ .
[ lim x n = ∞ , for
p + q ⋅ 0 + r ⋅ 0 +K+ z ⋅ 0
p
x →∞
€
n>0]
For the limit at negative infinity, the results for Cases I and II are unchanged.
The situation for Case III, however, is rather more complicated, since the parity of the
exponent n matters for the limit at negative infinity
lim x n = ∞ , for n > 0 and n even , but lim x n = − ∞ , for n > 0 and n odd .
x → −∞
x → −∞
So for
lim f (x) = lim
x →−∞
€
A) if
€
€
x → −∞
a x m + b x m−1 + K + k
, with m > n ,
p x n + q x€n−1 + K + z
a
p > 0 (meaning a and p have the same sign), and
1) the difference in degrees m – n is even,
then for large values of | x | , f(x) behaves like x to an even power, so
lim f (x) = + ∞
x → ±∞
, while if
2) the difference in degrees m – n is odd,
then for large values of | x | , f(x) behaves like x to an odd power, so
€
lim f (x) = + ∞ ,
x → +∞
lim f (x) = − ∞ ;
x → −∞
a
p < 0 (meaning a and p have opposite signs), the signs of the
€ limits are reversed, so we find that when
B) whereas if
1) the difference in degrees m – n is even,
€
2) the difference in degrees m – n is odd,
lim f (x) = − ∞ ,
x → +∞
lim f (x) = − ∞ , while if
x → ±∞
lim f (x) = + ∞ .
€
x → −∞
Graphs illustrating the limits at positive and negative infinity (called the asymptotic
behavior of the function f(x) ) for these various cases are presented on the next page.
€
Amendment –
One additional type of rational function we could look at is the kind where the
numerator or denominator contains an even root of a polynomial. We will choose as our
example the function used in Problems 23 and 24 of Section 2.6,
9x 6 − x
.
x3+ 1
f (x) =
When we are asked to evaluate the limit of such a function at positive infinity
(Problem 23), we can simply use the technique we have employed throughout these
notes. We will divide the numerator and denominator of€this ratio by x3 and use the
special “limit law” for limits at infinity:
9x 6
lim
−x
x3+ 1
x → +∞
= lim
x → +∞
9x 6
−x
x3+ 1
1
9x 6
3
⋅ x1 = lim
x → +∞
x3
−x
x3+ 1
6
9 x 6 − x6
1
6
⋅ x1 = lim
x3
x
x3
x3
x → +∞
9 − lim
€
=
x
6
x → +∞ x
lim 13
x
1+
9 − lim
=
x → +∞
1+
x
+
1
x3
1
5
x → +∞ x
lim 13
x
9−0
= 3 .
1+ 0
=
x → +∞
If we now wish to evaluate the limit at negative infinity (Problem 24), there is an
important detail we must keep in mind: the square root of x2 is not x , but rather | x |
€(
x 2 = x ) . What this means is that when x is a positive number, squaring x and
then taking the square root of the result gives us back x , but when x is negative,
squaring x and then taking the square root of that result loses the original negative
€
x2 = − x .
sign, so we now get back −x . So for negative values of x , we must write
This modifies the calculation we worked through above, yielding
9x 6 − x
= lim
x3+ 1
x → −∞
lim
x → −∞
1
9x 6 − x x 3
⋅ 1 = lim
x3+ 1
x → −∞
x3
− 9 − lim
€
=
1+
9x 6€− x
⋅
x3+ 1
− 16
x
1
x3
1
5
x → −∞ x
lim 13
x
= −
9−0
= −3 .
1+ 0
x → −∞
The effect of the even root is a “splitting” of the horizontal asymptote: while there is a
single horizontal asymptote for the rational functions covered in Case I above, we have
here two horizontal
asymptotes, one each for positive and negative values of x . Graphs
€
for this function and the function similar to the one used in Problem 22 are shown on
the next page. (Since odd roots of negative numbers are negative, there is no peculiar
behavior of the signs of the limits at positive and negative infinity in such cases.)
G. Ruffa
original handout – c. 2006
revised and graphs added – September 2008
amended for radicals – 26-27 August 2009