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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

Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
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
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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