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  • Calculate the limit of a function as x increases or decreases without bound.
  • Recognize a horizontal asymptote on the graph of a function.
  • Estimate the end behavior of a function as x increases or decreases without bound.
  • Recognize an oblique asymptote on the graph of a function.
  • Analyze a function and its derivatives to draw its graph.

We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x ± . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f .

Limits at infinity

We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs , we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.

Limits at infinity and horizontal asymptotes

Recall that lim x a f ( x ) = L means f ( x ) becomes arbitrarily close to L as long as x is sufficiently close to a . We can extend this idea to limits at infinity. For example, consider the function f ( x ) = 2 + 1 x . As can be seen graphically in [link] and numerically in [link] , as the values of x get larger, the values of f ( x ) approach 2 . We say the limit as x approaches of f ( x ) is 2 and write lim x f ( x ) = 2 . Similarly, for x < 0 , as the values | x | get larger, the values of f ( x ) approaches 2 . We say the limit as x approaches of f ( x ) is 2 and write lim x a f ( x ) = 2 .

The function f(x) 2 + 1/x is graphed. The function starts negative near y = 2 but then decreases to −∞ near x = 0. The function then decreases from ∞ near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.
The function approaches the asymptote y = 2 as x approaches ± .
Values of a function f As x ±
x 10 100 1,000 10,000
2 + 1 x 2.1 2.01 2.001 2.0001
x −10 −100 −1000 −10,000
2 + 1 x 1.9 1.99 1.999 1.9999

More generally, for any function f , we say the limit as x of f ( x ) is L if f ( x ) becomes arbitrarily close to L as long as x is sufficiently large. In that case, we write lim x a f ( x ) = L . Similarly, we say the limit as x of f ( x ) is L if f ( x ) becomes arbitrarily close to L as long as x < 0 and | x | is sufficiently large. In that case, we write lim x f ( x ) = L . We now look at the definition of a function having a limit at infinity.

Definition

(Informal) If the values of f ( x ) become arbitrarily close to L as x becomes sufficiently large, we say the function f has a limit at infinity    and write

lim x f ( x ) = L .

If the values of f ( x ) becomes arbitrarily close to L for x < 0 as | x | becomes sufficiently large, we say that the function f has a limit at negative infinity and write

lim x f ( x ) = L .

If the values f ( x ) are getting arbitrarily close to some finite value L as x or x , the graph of f approaches the line y = L . In that case, the line y = L is a horizontal asymptote of f ( [link] ). For example, for the function f ( x ) = 1 x , since lim x f ( x ) = 0 , the line y = 0 is a horizontal asymptote of f ( x ) = 1 x .

Definition

If lim x f ( x ) = L or lim x f ( x ) = L , we say the line y = L is a horizontal asymptote    of f .

The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.
(a) As x , the values of f are getting arbitrarily close to L . The line y = L is a horizontal asymptote of f . (b) As x , the values of f are getting arbitrarily close to M . The line y = M is a horizontal asymptote of f .

Questions & Answers

how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
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
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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
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
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
Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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