Advanced Placement Calculus
Limits and Continuity
Chapter 2
Section 2
Limits Involving Infinity
Essential Question: How does infinity affect the end behavior of a function?
Student Objectives: The student will determine the limit of a function as the x-values
approaches positive or negative infinity.
The student will determine the horizontal asymptote of a function,
The student will determine if the limit of a function as the x-value
approaches the value a is positive or negative infinity using
“fuzzy math”.
The student will apply the concepts of limits to determine the
end behavior model of a function.
Terms:
End Behavior Model
Horizontal Asymptotes
Sandwich Theorem
Vertical Asymptote
Equations:
Horizontal Asymptotes
The line y is a horizontal asymptotes of the function y = f(x) if either
lim f ( x ) = b OR
lim f ( x ) = b .
x→∞
x→−∞
Properties of Limits of Infinity
Given the following values :
lim f ( x ) = L
x→±∞
1.
2.
3.
4.
5.
6.
lim g ( x ) = M
and
x→±∞
lim ( f ( x ) + g ( x )) = L + M
x→±∞
lim ( f ( x ) − g ( x )) = L − M
x→±∞
lim ( f ( x ) ⋅ g ( x )) = LM
x→±∞
lim ( kf ( x )) = kL
x→±∞
⎛ f ( x)⎞ L
lim ⎜
= ,M ≠0
x→±∞ ⎝ g ( x ) ⎟
⎠ M
r
s
(
)
r
s
r
lim ( f ( x )) = lim f ( x ) = Ls
x→±∞
x→±∞
Vertical Asymptote
The line x = a is a vertical asymptote of the graph of the function y = f(x) if either lim− f ( x ) = ±∞ OR
lim+ f ( x ) = ±∞
x→a
x→a
End Behavior Model
The function g(x) is
a. A right end behavior model for f if and only if lim f ( x ) = 1 .
x→∞
b. A right end behavior model for f if and only if lim f ( x ) = 1 .
x→−∞
Graphing Calculator Skills:
Zoom “WAY OUT” for the domain values of a graph
Sample Questions:
3x 4 − 5x 2 + 7
x→∞ 8x 3 − 9x + 1
1. Determine the value of the following expression: lim
3x 3 + 4x − 2
x→∞ 8x 5 + 7x 3 − 5
2. Determine the value of the following expression: lim
4x 4 − 3x 2 − 9
3. Determine the value of the following expression: lim
x→∞ 5x 4 + 8x + 4
4. Determine the value of the following expression:
4x 3
4x 3
4x 3
lim
lim
and lim
2 ,
2
x→−3 x + 3
x→−3+ ( x + 3)
x→−3+ ( x + 3)
(
)2
5. Determine the end behavior for the following expression:
2x 7 − 6x 3 − 5x 2 − 2
lim
x→∞
5x 4 + 4x − 8
Homework:
Pages 76 - 77
Exercises 1, 5, 9, 19, 27, 29, 31, 35, 41, 49, and 55
Exercises: 2, 6, 12, 18, 24, 30, 32, 36, 42, 50, and 56