2.5 Limits Involving Infinity - Math TAMU
MATH 131: Lecture Notes
Summer II 2015
2.5 Limits Involving Infinity Recall that lim− f (x) or lim+ f (x) does not exist if we cannot make the values of f (x) arbitrarily x→a
x→a
close to a .......................... L. This happens when f increases or decreases without bound as x approaches to a point a. Preview Activity: Explain in words the behavior of the function as x approaches a for each graph below. Which symbol can we use if we want to express this behavior using the limit notation?
c
2015 Fatma Terzioglu
15
MATH 131: Lecture Notes
Summer II 2015
Activity 2: Graph each function to find the limit. Use −∞ or ∞ when appropriate. 1 x→0 x2
(a) lim
(b) lim+ ln x x→0
(c) lim log3 (x − 3) x→3
x4 − 4x2 − 7 (d) lim √ x2 − 3 x→ 3
Definition: The line x = a is called a .......................... asymptote of the curve y = f (x) if at least one of the following is true : lim f (x) = ∞
x→a
lim f (x) = −∞
x→a
c
2015 Fatma Terzioglu
lim f (x) = ∞
x→a−
lim f (x) = −∞
x→a−
16
lim f (x) = ∞
x→a+
lim f (x) = −∞
x→a+
MATH 131: Lecture Notes
Summer II 2015
Activity 1: Find the vertical asymptote(s) of the function f (x) =
x+2 . x2 + 5x + 6
Limits at Infinity In computing infinite limits, we let x approach a number and the result was that the values of f (x) became arbitrarily large or arbitrarily small. Here we let x become arbitrarily large or arbitrarily small and see what happens to f (x).
Preview Activity: Explain in words the end behavior of the function for each graph below. Which symbol can we use if we want to express this behavior using the limit notation?
c
2015 Fatma Terzioglu
17
MATH 131: Lecture Notes
Summer II 2015
Definition: • If the values of f (x) eventually get as close as we like to a number L as x increases without bound, then we write
• If the values of f (x) eventually get as close as we like to a number L as x decreases without bound, then we write
• If either limit holds, we call the line y = L a ...................... asymptote for the graph of f .
Activity 1: Find the horizontal asymptotes of f (x) =
c
2015 Fatma Terzioglu
18
1 . Are there any vertical asymptotes? x
MATH 131: Lecture Notes
Summer II 2015
Activity 2: Find the limits of the following. (a) lim ex x→−∞
3x2 − x − 2 x→∞ 5x2 + 4x + 1
(b) lim
t2 + 2 t→−∞ t3 + t2 − 1
(c) lim
c
2015 Fatma Terzioglu
19
MATH 131: Lecture Notes
Summer II 2015
x+2 (d) lim √ x→∞ 9x2 + 1
(e) lim
x→∞
√
x2 + 1 − x
(f) lim sin x x→∞
c
2015 Fatma Terzioglu
20
MATH 131: Lecture Notes
Summer II 2015
Infinite Limits at Infinity The notation lim f (x) = ∞
x→∞
is used to indicate that the values of f (x) become large as x becomes large. Similar meanings are attached to the following symbols:
Activity 3: Find the following limits. (a) lim ln x x→∞
(b) lim ex x→∞
(c) lim
x→∞
√
x2 + 1
(d) lim x3 − 3x x→∞
c
2015 Fatma Terzioglu
21
Summer II 2015
2.5 Limits Involving Infinity Recall that lim− f (x) or lim+ f (x) does not exist if we cannot make the values of f (x) arbitrarily x→a
x→a
close to a .......................... L. This happens when f increases or decreases without bound as x approaches to a point a. Preview Activity: Explain in words the behavior of the function as x approaches a for each graph below. Which symbol can we use if we want to express this behavior using the limit notation?
c
2015 Fatma Terzioglu
15
MATH 131: Lecture Notes
Summer II 2015
Activity 2: Graph each function to find the limit. Use −∞ or ∞ when appropriate. 1 x→0 x2
(a) lim
(b) lim+ ln x x→0
(c) lim log3 (x − 3) x→3
x4 − 4x2 − 7 (d) lim √ x2 − 3 x→ 3
Definition: The line x = a is called a .......................... asymptote of the curve y = f (x) if at least one of the following is true : lim f (x) = ∞
x→a
lim f (x) = −∞
x→a
c
2015 Fatma Terzioglu
lim f (x) = ∞
x→a−
lim f (x) = −∞
x→a−
16
lim f (x) = ∞
x→a+
lim f (x) = −∞
x→a+
MATH 131: Lecture Notes
Summer II 2015
Activity 1: Find the vertical asymptote(s) of the function f (x) =
x+2 . x2 + 5x + 6
Limits at Infinity In computing infinite limits, we let x approach a number and the result was that the values of f (x) became arbitrarily large or arbitrarily small. Here we let x become arbitrarily large or arbitrarily small and see what happens to f (x).
Preview Activity: Explain in words the end behavior of the function for each graph below. Which symbol can we use if we want to express this behavior using the limit notation?
c
2015 Fatma Terzioglu
17
MATH 131: Lecture Notes
Summer II 2015
Definition: • If the values of f (x) eventually get as close as we like to a number L as x increases without bound, then we write
• If the values of f (x) eventually get as close as we like to a number L as x decreases without bound, then we write
• If either limit holds, we call the line y = L a ...................... asymptote for the graph of f .
Activity 1: Find the horizontal asymptotes of f (x) =
c
2015 Fatma Terzioglu
18
1 . Are there any vertical asymptotes? x
MATH 131: Lecture Notes
Summer II 2015
Activity 2: Find the limits of the following. (a) lim ex x→−∞
3x2 − x − 2 x→∞ 5x2 + 4x + 1
(b) lim
t2 + 2 t→−∞ t3 + t2 − 1
(c) lim
c
2015 Fatma Terzioglu
19
MATH 131: Lecture Notes
Summer II 2015
x+2 (d) lim √ x→∞ 9x2 + 1
(e) lim
x→∞
√
x2 + 1 − x
(f) lim sin x x→∞
c
2015 Fatma Terzioglu
20
MATH 131: Lecture Notes
Summer II 2015
Infinite Limits at Infinity The notation lim f (x) = ∞
x→∞
is used to indicate that the values of f (x) become large as x becomes large. Similar meanings are attached to the following symbols:
Activity 3: Find the following limits. (a) lim ln x x→∞
(b) lim ex x→∞
(c) lim
x→∞
√
x2 + 1
(d) lim x3 − 3x x→∞
c
2015 Fatma Terzioglu
21
