Math 142 Lecture Notes for Section 5.3
Section 5.3 -
1
Limits at Infinity
Example 5.3.1:
Consider the functions x4 ,
1
x
and
|x|
as x approaches both positive and negative infinity.
x
Limits of Power Functions at infinity Suppose that p > 0 and k is any real number,
then
(1)
lim kxp =
x−
→∞
(2)
lim kxp =
x−
→−∞
(3)
k
lim p =
x−
→∞ x
(4)
k
lim
=
x−
→−∞ xp
Math 142 Lecture Notes for Section 5.3
2
Definition 5.3.2:
A line y = b is a horizontal asymptote of f (x) if
lim f (x) = b
x−
→∞
or
lim f (x) = b
x−
→−∞
Example 5.3.3:
Let g(x) = −4x4 + 2x3 − 3x2 + 5x + 1, find
(a) Find lim g(x)
x−
→∞
(b) Find
lim g(x)
x−
→−∞
Cheap and dirty tricks for the limits at infinity of rational functions. Find the highest
degree amongst the numerator and the denominator of r(x).
(1) If it is in the denominator of r(x), then
(2) If it is in the numerator of r(x), then
lim r(x) =
x−
→±∞
lim r(x) =
x−
→±∞
(3) if the degree of the numerator is the same as the degree of the denominator of r(x),
then lim r(x) =
x−
→±∞
Math 142 Lecture Notes for Section 5.3
Example 5.3.4:
Determine the following limits.
4x + 5
=
(a) lim 2
x−
→∞ x − 3x + 6
6x2 + 3x − 5
=
4x + 3
(b)
lim
x−
→−∞
(c)
3x2 + 5x
lim
=
x−
→−∞ 8x2 − 4
Example 5.3.5:
Find the vertical and horizontal asymptotes of the following functions:
(a) f (x) =
3x − 6
2x2 − 4
(b) g(x) =
3x + 5
4x − 3
3
Math 142 Lecture Notes for Section 5.3
(c) h(x) =
x2 − 4
x+2
(d) u(x) =
5ex
2 − 4ex
Section 5.3 Suggested Homework: 1-17(odd), 21, 27, 31
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