Limits to Infinity
In this handout we will examine limits of the form
g(x)
x→∞ h(x)
lim
g(x)
x→−∞ h(x)
or
lim
where g(x) and h(x) are polynomials.
The important thing to remark is that we are taking the limit to ∞ or −∞
of a rational function. We note that
Any finite number
=0
x→∞
xn
lim
Any finite number
=0
x→−∞
xn
or
lim
for n ∈ N.
Using the above facts, we try to rewrite our functions using terms like xcn
where c ∈ R. To do this we divide every term in the numerator and denominator by the highest power or x that appears in the function.
Example 1
x2 − 9x + 8
=
lim
x→∞
5x2 − x
=
lim
x2
x2
x→∞
lim
1
−
9x
x2
+
5x2
x2
− x9
−
x
x2
8
x2
x→∞
+
5 − x1
8
x2
.
We are now in a position to take the limit. We get
1 − x9 +
x→∞
5 − x1
lim
8
x2
1−0+0
5−0
1
=
.
5
=
In summary, we have found that
x2 − 9x + 8
1
= .
2
x→∞
5x − x
5
lim
Material developed by the Department of Mathematics & Statistics, N.U.I. Maynooth
and supported by the NDLR (www.ndlr.com).
1
Example 2
3x2 − 6x + 2
lim
=
x→−∞ 5x3 − 2x2 + 6x + 7
=
3x2
x3
lim
−
6x
x3
+
2
x3
3
2
x→−∞ 5x − 2x + 6x + 7
x3
x3
x3
x3
3
6
2
− x2 + x3
x
lim
x→−∞ 5 − 2 + 62 + 73
x
x
x
0−0+0
5−0+0+0
= 0.
=
Example 3
4x3 + 5
lim
=
x→∞ 5x2 − 1
=
4x3
x3
lim 5x
2
x→∞
x3
lim
x→∞
= ∞.
+
−
5
x3
1
x3
4 + x53
5
− x13
x
Try the following exercises for practice.
Calculate the following limits:
(a)
x4 − 2x2 − x
x→∞ 2x4 + 3x3 + 4
lim
(b)
3x3 − 2x2 − 2
x→∞ x5 − x2 + 1
lim
(c)
3x5 + 6x2
x→∞ 7x4 − 3x2 − 4
lim
(d)
3x6 + 4x5 − x3 − 2x2 − 1
x→−∞
5x6 − 4x3 + 3x + 2
lim
(e)
5x2 − 1
x→−∞ 3x6 − 2x2 − 7
lim
Solutions
(a) 12
(b) 0
(c) ∞
2
(d)
3
5
(e) 0