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Let’s consider several classes of functions here and look at the different types of end behaviors for these functions.

End behavior for polynomial functions

Consider the power function f ( x ) = x n where n is a positive integer. From [link] and [link] , we see that

lim x x n = ; n = 1 , 2 , 3 ,…

and

lim x x n = { ; n = 2 , 4 , 6 ,… ; n = 1 , 3 , 5 ,… .
The functions x2, x4, and x6 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.
For power functions with an even power of n , lim x x n = = lim x x n .
The functions x, x3, and x5 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.
For power functions with an odd power of n , lim x x n = and lim x x n = .

Using these facts, it is not difficult to evaluate lim x c x n and lim x c x n , where c is any constant and n is a positive integer. If c > 0 , the graph of y = c x n is a vertical stretch or compression of y = x n , and therefore

lim x c x n = lim x x n and lim x c x n = lim x x n if c > 0 .

If c < 0 , the graph of y = c x n is a vertical stretch or compression combined with a reflection about the x -axis, and therefore

lim x c x n = lim x x n and lim x c x n = lim x x n if c < 0 .

If c = 0 , y = c x n = 0 , in which case lim x c x n = 0 = lim x c x n .

Limits at infinity for power functions

For each function f , evaluate lim x f ( x ) and lim x f ( x ) .

  1. f ( x ) = −5 x 3
  2. f ( x ) = 2 x 4
  1. Since the coefficient of x 3 is −5 , the graph of f ( x ) = −5 x 3 involves a vertical stretch and reflection of the graph of y = x 3 about the x -axis. Therefore, lim x ( −5 x 3 ) = and lim x ( −5 x 3 ) = .
  2. Since the coefficient of x 4 is 2 , the graph of f ( x ) = 2 x 4 is a vertical stretch of the graph of y = x 4 . Therefore, lim x 2 x 4 = and lim x 2 x 4 = .
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Let f ( x ) = −3 x 4 . Find lim x f ( x ) .

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We now look at how the limits at infinity for power functions can be used to determine lim x ± f ( x ) for any polynomial function f . Consider a polynomial function

f ( x ) = a n x n + a n 1 x n 1 + + a 1 x + a 0

of degree n 1 so that a n 0 . Factoring, we see that

f ( x ) = a n x n ( 1 + a n 1 a n 1 x + + a 1 a n 1 x n 1 + a 0 a n ) .

As x ± , all the terms inside the parentheses approach zero except the first term. We conclude that

lim x ± f ( x ) = lim x ± a n x n .

For example, the function f ( x ) = 5 x 3 3 x 2 + 4 behaves like g ( x ) = 5 x 3 as x ± as shown in [link] and [link] .

Both functions f(x) = 5x3 – 3x2 + 4 and g(x) = 5x3 are plotted. Their behavior for large positive and large negative numbers converges.
The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.
A polynomial’s end behavior is determined by the term with the largest exponent.
x 10 100 1000
f ( x ) = 5 x 3 3 x 2 + 4 4704 4,970,004 4,997,000,004
g ( x ) = 5 x 3 5000 5,000,000 5,000,000,000
x −10 −100 −1000
f ( x ) = 5 x 3 3 x 2 + 4 −5296 −5,029,996 −5,002,999,996
g ( x ) = 5 x 3 −5000 −5,000,000 −5,000,000,000

End behavior for algebraic functions

The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In [link] , we show that the limits at infinity of a rational function f ( x ) = p ( x ) q ( x ) depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of x appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of x .

Determining end behavior for rational functions

For each of the following functions, determine the limits as x and x . Then, use this information to describe the end behavior of the function.

  1. f ( x ) = 3 x 1 2 x + 5 (Note: The degree of the numerator and the denominator are the same.)
  2. f ( x ) = 3 x 2 + 2 x 4 x 3 5 x + 7 (Note: The degree of numerator is less than the degree of the denominator.)
  3. f ( x ) = 3 x 2 + 4 x x + 2 (Note: The degree of numerator is greater than the degree of the denominator.)
  1. The highest power of x in the denominator is x . Therefore, dividing the numerator and denominator by x and applying the algebraic limit laws, we see that
    lim x ± 3 x 1 2 x + 5 = lim x ± 3 1 / x 2 + 5 / x = lim x ± ( 3 1 / x ) lim x ± ( 2 + 5 / x ) = lim x ± 3 lim x ± 1 / x lim x ± 2 + lim x ± 5 / x = 3 0 2 + 0 = 3 2 .

    Since lim x ± f ( x ) = 3 2 , we know that y = 3 2 is a horizontal asymptote for this function as shown in the following graph.
    The function f(x) = (3x + 1)/(2x + 5) is plotted as is its horizontal asymptote at y = 3/2.
    The graph of this rational function approaches a horizontal asymptote as x ± .
  2. Since the largest power of x appearing in the denominator is x 3 , divide the numerator and denominator by x 3 . After doing so and applying algebraic limit laws, we obtain
    lim x ± 3 x 2 + 2 x 4 x 3 5 x + 7 = lim x ± 3 / x + 2 / x 2 4 5 / x 2 + 7 / x 3 = 3.0 + 2.0 4 5.0 + 7.0 = 0 .

    Therefore f has a horizontal asymptote of y = 0 as shown in the following graph.
    The function f(x) = (3x2 + 2x)/(4x2 – 5x + 7) is plotted as is its horizontal asymptote at y = 0.
    The graph of this rational function approaches the horizontal asymptote y = 0 as x ± .
  3. Dividing the numerator and denominator by x , we have
    lim x ± 3 x 2 + 4 x x + 2 = lim x ± 3 x + 4 1 + 2 / x .

    As x ± , the denominator approaches 1 . As x , the numerator approaches + . As x , the numerator approaches . Therefore lim x f ( x ) = , whereas lim x f ( x ) = as shown in the following figure.
    The function f(x) = (3x2 + 4x)/(x + 2) is plotted. It appears to have a diagonal asymptote as well as a vertical asymptote at x = −2.
    As x , the values f ( x ) . As x , the values f ( x ) .
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Questions & Answers

can you give me a problem for function. a trigonometric one
geovanni Reply
state and prove L hospital rule
Krishna Reply
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
Jawad
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t=0), total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t . Interpret the slope of the graph of S . Use part a. to predict the year when online shopping during Christmas will reach $60 billion?
Nguyen Reply
what is the derivative of x= Arc sin (x)^1/2
morfling Reply
y^2 = arcsin(x)
Pitior
x = sin (y^2)
Pitior
differentiate implicitly
Pitior
then solve for dy/dx
Pitior
thank you it was very helpful
morfling
questions solve y=sin x
Obi Reply
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
follow algebraic method. look under factoring numerator from Khan academy
moe
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
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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