<< Chapter < Page Chapter >> Page >

Definition

(Informal) We say a function f has an infinite limit at infinity and write

lim x f ( x ) = .

if f ( x ) becomes arbitrarily large for x sufficiently large. We say a function has a negative infinite limit at infinity and write

lim x f ( x ) = .

if f ( x ) < 0 and | f ( x ) | becomes arbitrarily large for x sufficiently large. Similarly, we can define infinite limits as x .

Formal definitions

Earlier, we used the terms arbitrarily close , arbitrarily large , and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.

Definition

(Formal) We say a function f has a limit at infinity    , if there exists a real number L such that for all ε > 0 , there exists N > 0 such that

| f ( x ) L | < ε

for all x > N . In that case, we write

lim x f ( x ) = L

(see [link] ).

We say a function f has a limit at negative infinity if there exists a real number L such that for all ε > 0 , there exists N < 0 such that

| f ( x ) L | < ε

for all x < N . In that case, we write

lim x f ( x ) = L .
The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + ॉ and L – ॉ. On the x axis, N is marked as the value of x such that f(x) = L + ॉ.
For a function with a limit at infinity, for all x > N , | f ( x ) L | < ε .

Earlier in this section, we used graphical evidence in [link] and numerical evidence in [link] to conclude that lim x ( 2 + 1 x ) = 2 . Here we use the formal definition of limit at infinity to prove this result rigorously.

Use the formal definition of limit at infinity to prove that lim x ( 2 + 1 x ) = 2 .

Let ε > 0 . Let N = 1 ε . Therefore, for all x > N , we have

| 2 + 1 x 2 | = | 1 x | = 1 x < 1 N = ε
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the formal definition of limit at infinity to prove that lim x ( 3 1 x 2 ) = 3 .

Let ε > 0 . Let N = 1 ε . Therefore, for all x > N , we have

| 3 1 x 2 3 | = 1 x 2 < 1 N 2 = ε

Therefore, lim x ( 3 1 / x 2 ) = 3 .

Got questions? Get instant answers now!

We now turn our attention to a more precise definition for an infinite limit at infinity.

Definition

(Formal) We say a function f has an infinite limit at infinity    and write

lim x f ( x ) =

if for all M > 0 , there exists an N > 0 such that

f ( x ) > M

for all x > N (see [link] ).

We say a function has a negative infinite limit at infinity and write

lim x f ( x ) =

if for all M < 0 , there exists an N > 0 such that

f ( x ) < M

for all x > N .

Similarly we can define limits as x .

The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.
For a function with an infinite limit at infinity, for all x > N , f ( x ) > M .

Earlier, we used graphical evidence ( [link] ) and numerical evidence ( [link] ) to conclude that lim x x 3 = . Here we use the formal definition of infinite limit at infinity to prove that result.

Use the formal definition of infinite limit at infinity to prove that lim x x 3 = .

Let M > 0 . Let N = M 3 . Then, for all x > N , we have

x 3 > N 3 = ( M 3 ) 3 = M .

Therefore, lim x x 3 = .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the formal definition of infinite limit at infinity to prove that lim x 3 x 2 = .

Let M > 0 . Let N = M 3 . Then, for all x > N , we have

3 x 2 > 3 N 2 = 3 ( M 3 ) 2 2 = 3 M 3 = M

Got questions? Get instant answers now!

End behavior

The behavior of a function as x ± is called the function’s end behavior    . At each of the function’s ends, the function could exhibit one of the following types of behavior:

  1. The function f ( x ) approaches a horizontal asymptote y = L .
  2. The function f ( x ) or f ( x ) .
  3. The function does not approach a finite limit, nor does it approach or . In this case, the function may have some oscillatory behavior.

Questions & Answers

A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
I can
Debdoot
Practice Key Terms 5

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask