Transcript 3 5 Limits Infinity

3-5: Limits at Infinity
Objectives:
Assignment:
1. To determine finite or • P. 205-208: 1, 2, 9, 11,
infinite limits at
21-33 odd, 41, 105,
infinity
106
2. To find horizontal
asymptotes of the
graph of a function
• P. 205-207: 3-8, 55,
59, 71, 87, 88
Warm-Up 1
For 𝑓 𝑥 = 3𝑥 2 , as 𝑥 → ∞, what does 𝑓 𝑥
approach?
𝑥
𝒇 𝒙
1
3
10
300
100
30000
1000
3000000
10000 300000000
Warm-Up 2
For 𝑓 𝑥 = 3𝑥 2 , as 𝑥 → −∞, what does 𝑓 𝑥
approach?
𝑥
𝒇 𝒙
1
3
10
300
100
30000
1000
3000000
10000 300000000
This is an
infinite limit
at infinity
that refers to
the end
behavior of
𝑓(𝑥).
Warm-Up 3
For 𝑓 𝑥 =
approach?
𝑥
3𝑥 2
,
𝑥 2 +1
as 𝑥 → ∞, what does 𝑓 𝑥
𝒇 𝒙
1
1.5
10
2.9703
100
2.9997
1000
2.999997
10000 2.99999997
This is a
finite limit at
infinity that
refers to the
horizontal
asymptote of
𝑓(𝑥).
Objective 1
You will be able to determine
finite or infinite limits at infinity
Limits at Infinity
3𝑥 2
,
𝑥 2 +1
For 𝑓 𝑥 =
as 𝑥 → ∞, 𝑓 𝑥 approaches
3. In limit notation:
3𝑥 2
lim
=3
𝑥→∞ 𝑥 2 + 1
Limit at infinity
Limits at Infinity
lim 𝑓 𝑥 = 𝐿
𝑥→±∞
This finite limit
at infinity means
that 𝑓 𝑥
approaches the
unique number 𝐿
as 𝑥 increases or
decreases
without bound.
Exercise 1
Find
5
lim
𝑥→∞ 𝑥 3
numerically.
𝒙
1
10
100
1000
𝒇 𝒙
5
.005
5 × 10−6
5 × 10−9
Limits at Infinity
If 𝑟 is a rational number and 𝑐 is any real
number, then
𝑐
lim 𝑟 = 0
𝑥→∞ 𝑥
and
𝑐
lim 𝑟 = 0
𝑥→−∞ 𝑥
If 𝑥^𝑟 is defined when 𝑥 < 0.
Exercise 2
Find the limit lim 5 −
𝑥→∞
2
.
𝑥2
Exercise 3
Find the limit
2𝑥−1
lim
.
𝑥→∞ 𝑥+1
2𝑥 − 1 → ∞
∞
lim
=
𝑥→∞ 𝑥 + 1 → ∞
∞
Indeterminate Form
Protip:
Divide top and
bottom by the
highest power in
the denominator.
You will be able to find horizontal
asymptotes of the graph of a function
Objective 2
Asymptotes
Horizontal Asymptote:
The line 𝑦 = 𝑏 is a horizontal asymptote of the
graph of 𝑓(𝑥) if 𝑓(𝑥) → 𝑏 as 𝑥 → +∞ or 𝑥 → −∞.
Horizontal Asymptote
The line 𝑦 = 𝐿 is a horizontal asymptote of
the graph of 𝑓(𝑥) if
lim 𝑓(𝑥) = 𝐿
𝑥→∞
or
lim 𝑓(𝑥) = 𝐿
𝑥→−∞
𝑓(𝑥) can have at most 2 horizontal asymptotes.
Exercise 4
Find each limit.
1.
2𝑥+5
lim 3𝑥 2+1
𝑥→∞
2.
2𝑥 2 +5
lim
𝑥→∞ 3𝑥 2 +1
3.
2𝑥 3 +5
lim
𝑥→∞ 3𝑥 2 +1
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
In other words, the 𝑥-axis is a
horizontal asymptote when the
degree of the denominator is
greater than the numerator.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
This happens because the
denominator is increasing faster
than the numerator. Thus the
function approaches zero.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
If d = n, then
𝑦 = 𝑎𝑏𝑛𝑛 is a
horizontal
asymptote.
In other words,
when the degrees
are equal, the
function approaches
the ratio of the
leading coefficients.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
If d = n, then
𝑦 = 𝑎𝑏𝑛𝑛 is a
horizontal
asymptote.
This happens
because the
numerator and
denominator are
increasing/decreasing
at roughly the same
rate.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
If d = n, then
𝑦 = 𝑎𝑏𝑛𝑛 is a
horizontal
asymptote.
If d < n, then
there are no
horizontal
asymptotes.
This happens because the
numerator is increasing
faster than the
denominator.
Exercise 5
Find each limit.
1.
lim
3𝑥−2
𝑥→∞ 2𝑥 2 +1
For 𝑥 > 0, 𝑥 = 𝑥 2
2.
lim
3𝑥−2
𝑥→−∞ 2𝑥 2 +1
For 𝑥 < 0, 𝑥 = − 𝑥 2
Exercise 6
Find each limit.
1.
lim sin 𝑥
𝑥→∞
2.
sin 𝑥
𝑥→∞ 𝑥
lim
Exercise 7
Find each limit.
1.
lim 𝑥 3
𝑥→∞
2.
lim 𝑥 3
𝑥→−∞
Infinite Limits at Infinity
Let f be a function defined on the interval
𝑎, ∞ .
The statement
lim 𝑓 𝑥 = ∞ means that
𝑥→∞
for each positive number
𝑀, there is a
corresponding number
𝑁 > 0 such that 𝑓 𝑥 > 𝑀
whenever 𝑥 > 𝑁.
The statement
lim 𝑓 𝑥 = −∞ means
𝑥→∞
that for each positive
number 𝑀, there is a
corresponding number
𝑁 > 0 such that 𝑓 𝑥 < 𝑀
whenever 𝑥 > 𝑁.
Infinite Limits at Infinity
Let f be a function defined on the interval
𝑎, ∞ .
lim 𝑥 3 = ∞ means that
The statement
lim 𝑓 𝑥 = ∞ means that
𝑥→∞
for each positive number
𝑀, there is a
corresponding number
𝑁 > 0 such that 𝑓 𝑥 > 𝑀
whenever 𝑥 > 𝑁.
𝑥→∞
if, say 𝑀 = 1000, there
is an 𝑁 such that 𝑥 3 >
1000 whenever 𝑥 > 𝑁.
𝑁 = 10
Exercise 8
Find each limit.
1.
2𝑥 2 −4𝑥
lim
𝑥→∞ 𝑥+1
2.
2𝑥 2 −4𝑥
lim
𝑥→−∞ 𝑥+1
3-5: Limits at Infinity
Objectives:
Assignment:
1. To determine finite or • P. 205-208: 1, 2, 9, 11,
infinite limits at
21-33 odd, 41, 105,
infinity
106
2. To find horizontal
asymptotes of the
graph of a function
• P. 205-207: 3-8, 55,
59, 71, 87, 88