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Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Lesson: Limits at Infinity and Asymptotes
Course Developer: Brijendra Yadav
Department/College: Assistant Professor, Department of
Mathematics, A.N.D. College, University of Delhi
Page 2
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Lesson: Limits at Infinity and Asymptotes
Course Developer: Brijendra Yadav
Department/College: Assistant Professor, Department of
Mathematics, A.N.D. College, University of Delhi
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Table of Contents
Chapter: Limits at Infinity and Asymptotes
• 1: Learning Outcomes
• 2: Introduction
• 3: Limits at Infinity
• 4: Infinite Limits at Infinity
• 5: Limits of Rational Functions at Infinity
• 6: Asymptotes
o 6.1. Horizontal Asymptotes
o 6.2. Vertical Asymptotes
o 6.3. Oblique Asymptotes
o 6.4. Dominant Term of a Function
• Exercises
• Summary
• Reference
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Limits at Infinity
? Infinite Limits at Infinity
? Limits of Rational Functions at Infinity
? Asymptotes
? Horizontal Asymptotes
? Vertical Asymptotes
? Oblique Asymptotes
? Dominant Term of a Function
Page 3
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Lesson: Limits at Infinity and Asymptotes
Course Developer: Brijendra Yadav
Department/College: Assistant Professor, Department of
Mathematics, A.N.D. College, University of Delhi
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Table of Contents
Chapter: Limits at Infinity and Asymptotes
• 1: Learning Outcomes
• 2: Introduction
• 3: Limits at Infinity
• 4: Infinite Limits at Infinity
• 5: Limits of Rational Functions at Infinity
• 6: Asymptotes
o 6.1. Horizontal Asymptotes
o 6.2. Vertical Asymptotes
o 6.3. Oblique Asymptotes
o 6.4. Dominant Term of a Function
• Exercises
• Summary
• Reference
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Limits at Infinity
? Infinite Limits at Infinity
? Limits of Rational Functions at Infinity
? Asymptotes
? Horizontal Asymptotes
? Vertical Asymptotes
? Oblique Asymptotes
? Dominant Term of a Function
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
2. Introduction:
Asymptotes convey the information about the behavior of curves
and determining the asymptotes of a function is an important step in
sketching its graph. In this lesson, we will study about the limits at
infinity and how to use the concept of limits to find the asymptote of a
function.
3: Limits at Infinity:
Let X be any non-empty subset of R, Let : X R f ? . Suppose that
00
(, ) X x for some x R 8? ? . The function f(x) is said to have a limit R ? ? as
x ? 8 if given any 0 e > there exists
0
() x d de = > such that for any x d > ,
then
() fx e -< ?
Limit at infinity is also denoted by
lim ( )
x
fx
?8
= ? .
Value Addition: Note
Limits at infinity i.e. the limits of f(x) as x ? ±8 are unique whenever they
exist.
4: Infinite Limits at Infinity:
Let X is a non-empty subset of R and let : X R f ? . Suppose that
00
(, ) X x for x R 8? ? . The function f(x) is said to tends to
() or as x +8 - 8 ? 8 if given any MR ? there exists a number
0
(M) x dd = >
such that for any x d > the function ( ) () () f x M or f x M respectively >< .
Page 4
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Lesson: Limits at Infinity and Asymptotes
Course Developer: Brijendra Yadav
Department/College: Assistant Professor, Department of
Mathematics, A.N.D. College, University of Delhi
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Table of Contents
Chapter: Limits at Infinity and Asymptotes
• 1: Learning Outcomes
• 2: Introduction
• 3: Limits at Infinity
• 4: Infinite Limits at Infinity
• 5: Limits of Rational Functions at Infinity
• 6: Asymptotes
o 6.1. Horizontal Asymptotes
o 6.2. Vertical Asymptotes
o 6.3. Oblique Asymptotes
o 6.4. Dominant Term of a Function
• Exercises
• Summary
• Reference
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Limits at Infinity
? Infinite Limits at Infinity
? Limits of Rational Functions at Infinity
? Asymptotes
? Horizontal Asymptotes
? Vertical Asymptotes
? Oblique Asymptotes
? Dominant Term of a Function
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
2. Introduction:
Asymptotes convey the information about the behavior of curves
and determining the asymptotes of a function is an important step in
sketching its graph. In this lesson, we will study about the limits at
infinity and how to use the concept of limits to find the asymptote of a
function.
3: Limits at Infinity:
Let X be any non-empty subset of R, Let : X R f ? . Suppose that
00
(, ) X x for some x R 8? ? . The function f(x) is said to have a limit R ? ? as
x ? 8 if given any 0 e > there exists
0
() x d de = > such that for any x d > ,
then
() fx e -< ?
Limit at infinity is also denoted by
lim ( )
x
fx
?8
= ? .
Value Addition: Note
Limits at infinity i.e. the limits of f(x) as x ? ±8 are unique whenever they
exist.
4: Infinite Limits at Infinity:
Let X is a non-empty subset of R and let : X R f ? . Suppose that
00
(, ) X x for x R 8? ? . The function f(x) is said to tends to
() or as x +8 - 8 ? 8 if given any MR ? there exists a number
0
(M) x dd = >
such that for any x d > the function ( ) () () f x M or f x M respectively >< .
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
The infinite limit at infinity is denoted by
lim ( ) lim ( )
xx
fx or fx
?8 ?8
= 8 = -8 .
Theorem 1: Let X be a non-empty subset of R and let f(x) and g(x) are
two functions defined on X and suppose that
00
(, ) X x for some x R 8? ? . And
suppose that
0
() 0 g x for all x x > > and that for some ,0 R ?? ?? , we have
()
lim
()
x
fx
gx
?8
= ? .
(I) If 0, lim ( ) lim ( )
xx
then f x if and only if g x
?8 ?8
> =8 =8 ? .
(II) If 0, lim ( ) lim ( )
xx
then f x if and only if g x
?8 ?8
< = -8 = 8 ? .
Proof: Let X be a non-empty subset of R such that
00
(, ) X x for some x R 8? ? and let f(x) and g(x) are two functions defined on
X and
0
() 0 g x for all x x > > and that for some ,0 R ?? ?? , we have
()
lim
()
x
fx
gx
?8
= ? .
Case (I): Let 0, > ? then there exists
10
xx > such that
1
1 () 3
0
2 () 2
fx
for all x x
gx
<< < > ??
?
1
13
() () ()
22
g x f x g x for all x x
? ? ? ?
<< >
? ? ? ?
? ? ? ?
?? (1)
thus from equation (1) it follows that
lim ( ) lim ( )
xx
f x if and only if g x
?8 ?8
=8=8
Case (II): Let 0, < ? then there exists a positive number k > 0 such that k = - ?
thus there exists
20
xx > such that
2
3 () 1
2 () 2
fx
k k for all x x
gx
- < <- >
?
2
31
() () ()
2 2
k g x f x k g x for all x x
?? ??
- < <- >
?? ??
?? ??
(2)
Page 5
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Lesson: Limits at Infinity and Asymptotes
Course Developer: Brijendra Yadav
Department/College: Assistant Professor, Department of
Mathematics, A.N.D. College, University of Delhi
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
Table of Contents
Chapter: Limits at Infinity and Asymptotes
• 1: Learning Outcomes
• 2: Introduction
• 3: Limits at Infinity
• 4: Infinite Limits at Infinity
• 5: Limits of Rational Functions at Infinity
• 6: Asymptotes
o 6.1. Horizontal Asymptotes
o 6.2. Vertical Asymptotes
o 6.3. Oblique Asymptotes
o 6.4. Dominant Term of a Function
• Exercises
• Summary
• Reference
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Limits at Infinity
? Infinite Limits at Infinity
? Limits of Rational Functions at Infinity
? Asymptotes
? Horizontal Asymptotes
? Vertical Asymptotes
? Oblique Asymptotes
? Dominant Term of a Function
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
2. Introduction:
Asymptotes convey the information about the behavior of curves
and determining the asymptotes of a function is an important step in
sketching its graph. In this lesson, we will study about the limits at
infinity and how to use the concept of limits to find the asymptote of a
function.
3: Limits at Infinity:
Let X be any non-empty subset of R, Let : X R f ? . Suppose that
00
(, ) X x for some x R 8? ? . The function f(x) is said to have a limit R ? ? as
x ? 8 if given any 0 e > there exists
0
() x d de = > such that for any x d > ,
then
() fx e -< ?
Limit at infinity is also denoted by
lim ( )
x
fx
?8
= ? .
Value Addition: Note
Limits at infinity i.e. the limits of f(x) as x ? ±8 are unique whenever they
exist.
4: Infinite Limits at Infinity:
Let X is a non-empty subset of R and let : X R f ? . Suppose that
00
(, ) X x for x R 8? ? . The function f(x) is said to tends to
() or as x +8 - 8 ? 8 if given any MR ? there exists a number
0
(M) x dd = >
such that for any x d > the function ( ) () () f x M or f x M respectively >< .
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
The infinite limit at infinity is denoted by
lim ( ) lim ( )
xx
fx or fx
?8 ?8
= 8 = -8 .
Theorem 1: Let X be a non-empty subset of R and let f(x) and g(x) are
two functions defined on X and suppose that
00
(, ) X x for some x R 8? ? . And
suppose that
0
() 0 g x for all x x > > and that for some ,0 R ?? ?? , we have
()
lim
()
x
fx
gx
?8
= ? .
(I) If 0, lim ( ) lim ( )
xx
then f x if and only if g x
?8 ?8
> =8 =8 ? .
(II) If 0, lim ( ) lim ( )
xx
then f x if and only if g x
?8 ?8
< = -8 = 8 ? .
Proof: Let X be a non-empty subset of R such that
00
(, ) X x for some x R 8? ? and let f(x) and g(x) are two functions defined on
X and
0
() 0 g x for all x x > > and that for some ,0 R ?? ?? , we have
()
lim
()
x
fx
gx
?8
= ? .
Case (I): Let 0, > ? then there exists
10
xx > such that
1
1 () 3
0
2 () 2
fx
for all x x
gx
<< < > ??
?
1
13
() () ()
22
g x f x g x for all x x
? ? ? ?
<< >
? ? ? ?
? ? ? ?
?? (1)
thus from equation (1) it follows that
lim ( ) lim ( )
xx
f x if and only if g x
?8 ?8
=8=8
Case (II): Let 0, < ? then there exists a positive number k > 0 such that k = - ?
thus there exists
20
xx > such that
2
3 () 1
2 () 2
fx
k k for all x x
gx
- < <- >
?
2
31
() () ()
2 2
k g x f x k g x for all x x
?? ??
- < <- >
?? ??
?? ??
(2)
Limits at Infinity and Asymptotes
Institute of Lifelong Learning, University of Delhi
thus from equation (2) it follows that
lim ( ) lim ( )
xx
f x if and only if g x
?8 ?8
= -8 = 8 .
Example 1: Show that lim
n
x
x for n N
?8
=8? .
Solution: Let ( ) (0, )
n
f x x for x = ?8
Given MR ? , let sup{ 1, } M d =
Then for all x d > , we have
()
n
fx x x M = =>
Since MR ? is arbitrary, it follows that
lim ( ) lim( )
n
xx
fx x
?8 ?8
= = 8.
Example 2: Prove that
( )
1/
lim 0
k
x
x for k
?8
=8> .
Solution: Let
( )
1/
()
k
fx x =
Given any MR ? , let
k
M d =
Now if x d > , then we have
1/ 1/ kk
x M d ==
Thus, it follows that
1/
lim( )
k
x
x
?8
= 8 .
Example 3: Prove that lim ( 1)
x
x
a for a
?8
=8> .
Solution: Let for 0 a > ,
1 a a = + [since a > 1]
?
[] xx
aa = [where [x] denotes integral value of x]
?
[]
(1 )
x
a >+
?
[]
1
x
a >+
?
[] x
a >
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FAQs on Lecture 3 - Limits at Infinity and Asymptotes - Calculus - Engineering Mathematics
| 1. What is the concept of limits at infinity in engineering mathematics? |  |
Ans. In engineering mathematics, limits at infinity refer to the behavior of a function as the input approaches positive or negative infinity. It determines the long-term trend or asymptotic behavior of the function.
| 2. How do you evaluate limits at infinity? | |
Ans. To evaluate limits at infinity, we examine the highest power of the variable in the function. If the highest power is positive, the function tends to positive or negative infinity depending on the sign of the coefficient. If the highest power is negative, the function tends to zero as the variable approaches infinity.
| 3. What are horizontal asymptotes in engineering mathematics? | |
Ans. Horizontal asymptotes are straight lines that a function approaches as the input approaches positive or negative infinity. They represent the long-term behavior of the function. If a function has a horizontal asymptote at y = c, where c is a constant, the function approaches the line y = c as the input approaches infinity or negative infinity.
| 4. Can a function have more than one horizontal asymptote? | |
Ans. No, a function can have at most one horizontal asymptote. It is possible for a function to have no horizontal asymptote or a single horizontal asymptote. If a function has multiple horizontal asymptotes, it violates the definition of a horizontal asymptote.
| 5. How do you determine the existence of a vertical asymptote in engineering mathematics? | |
Ans. To determine the existence of a vertical asymptote, we examine the behavior of the function as the input approaches a particular value. If the function approaches positive or negative infinity as the input approaches the value, then there is a vertical asymptote at that value. If the function approaches a finite number, there is no vertical asymptote at that value.
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