<< Chapter < Page Chapter >> Page >

Consider the function f ( x ) = 5 x 2 / 3 . Determine the point on the graph where a cusp is located. Determine the end behavior of f .

The function f has a cusp at ( 0 , 5 ) lim x 0 f ( x ) = , lim x 0 + f ( x ) = . For end behavior, lim x ± f ( x ) = .

Got questions? Get instant answers now!

Key concepts

  • The limit of f ( x ) is L as x (or as x ) if the values f ( x ) become arbitrarily close to L as x becomes sufficiently large.
  • The limit of f ( x ) is as x if f ( x ) becomes arbitrarily large as x becomes sufficiently large. The limit of f ( x ) is as x if f ( x ) < 0 and | f ( x ) | becomes arbitrarily large as x becomes sufficiently large. We can define the limit of f ( x ) as x approaches similarly.
  • For a polynomial function p ( x ) = a n x n + a n 1 x n 1 + + a 1 x + a 0 , where a n 0 , the end behavior is determined by the leading term a n x n . If n 0 , p ( x ) approaches or at each end.
  • For a rational function f ( x ) = p ( x ) q ( x ) , the end behavior is determined by the relationship between the degree of p and the degree of q . If the degree of p is less than the degree of q , the line y = 0 is a horizontal asymptote for f . If the degree of p is equal to the degree of q , then the line y = a n b n is a horizontal asymptote, where a n and b n are the leading coefficients of p and q , respectively. If the degree of p is greater than the degree of q , then f approaches or at each end.

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

For the following functions f ( x ) , determine whether there is an asymptote at x = a . Justify your answer without graphing on a calculator.

f ( x ) = x + 1 x 2 + 5 x + 4 , a = −1

Got questions? Get instant answers now!

f ( x ) = x x 2 , a = 2

Yes, there is a vertical asymptote

Got questions? Get instant answers now!

f ( x ) = ( x + 2 ) 3 / 2 , a = −2

Got questions? Get instant answers now!

f ( x ) = ( x 1 ) −1 / 3 , a = 1

Yes, there is vertical asymptote

Got questions? Get instant answers now!

f ( x ) = 1 + x −2 / 5 , a = 1

Got questions? Get instant answers now!

For the following exercises, evaluate the limit.

lim x 1 3 x + 6

0

Got questions? Get instant answers now!

lim x 2 x 5 4 x

Got questions? Get instant answers now!

lim x x 2 2 x + 5 x + 2

Got questions? Get instant answers now!

lim x 3 x 3 2 x x 2 + 2 x + 8

Got questions? Get instant answers now!

lim x x 4 4 x 3 + 1 2 2 x 2 7 x 4

1 7

Got questions? Get instant answers now!

lim x 3 x x 2 + 1

Got questions? Get instant answers now!

lim x 4 x 2 1 x + 2

−2

Got questions? Get instant answers now!

lim x 4 x x 2 1

Got questions? Get instant answers now!

lim x 4 x x 2 1

−4

Got questions? Get instant answers now!

lim x 2 x x x + 1

Got questions? Get instant answers now!

For the following exercises, find the horizontal and vertical asymptotes.

f ( x ) = x 9 x

Horizontal: none, vertical: x = 0

Got questions? Get instant answers now!

f ( x ) = x 3 4 x 2

Horizontal: none, vertical: x = ± 2

Got questions? Get instant answers now!

f ( x ) = x 2 + 3 x 2 + 1

Got questions? Get instant answers now!

f ( x ) = sin ( x ) sin ( 2 x )

Horizontal: none, vertical: none

Got questions? Get instant answers now!

f ( x ) = cos x + cos ( 3 x ) + cos ( 5 x )

Got questions? Get instant answers now!

f ( x ) = x sin ( x ) x 2 1

Horizontal: y = 0 , vertical: x = ± 1

Got questions? Get instant answers now!

f ( x ) = 1 x 3 + x 2

Horizontal: y = 0 , vertical: x = 0 and x = −1

Got questions? Get instant answers now!

f ( x ) = 1 x 1 2 x

Got questions? Get instant answers now!

f ( x ) = x 3 + 1 x 3 1

Horizontal: y = 1 , vertical: x = 1

Got questions? Get instant answers now!

f ( x ) = sin x + cos x sin x cos x

Got questions? Get instant answers now!

f ( x ) = x sin x

Horizontal: none, vertical: none

Got questions? Get instant answers now!

For the following exercises, construct a function f ( x ) that has the given asymptotes.

x = 1 and y = 2

Answers will vary, for example: y = 2 x x 1

Got questions? Get instant answers now!

y = 4 , x = −1

Answers will vary, for example: y = 4 x x + 1

Got questions? Get instant answers now!

For the following exercises, graph the function on a graphing calculator on the window x = [ −5 , 5 ] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.

[T] f ( x ) = 1 x + 10

y = 0

Got questions? Get instant answers now!

[T] f ( x ) = x + 1 x 2 + 7 x + 6

Got questions? Get instant answers now!

[T] lim x x 2 + 10 x + 25

Got questions? Get instant answers now!

[T] lim x x + 2 x 2 + 7 x + 6

Got questions? Get instant answers now!

[T] lim x 3 x + 2 x + 5

y = 3

Got questions? Get instant answers now!

For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.

y = 2 x + 1 x 2 + 6 x + 5

Got questions? Get instant answers now!

y = x 3 + 4 x 2 + 3 x 3 x + 9


An upward-facing parabola with minimum between x = 0 and x = −1 with y intercept between 0 and 1.

Got questions? Get instant answers now!

y = x 2 + x 2 x 2 3 x 4

Got questions? Get instant answers now!

y = cos x x , on x = [ −2 π , 2 π ]


This graph has vertical asymptote at x = 0. The first part of the function occurs in the second and third quadrants and starts in the third quadrant just below (−2π, 0), increases and passes through the x axis at −3π/2, reaches a maximum and then decreases through the x axis at −π/2 before approaching the asymptote. On the other side of the asymptote, the function starts in the first quadrant, decreases quickly to pass through π/2, decreases to a local minimum and then increases through (3π/2, 0) before staying just above (2π, 0).

Got questions? Get instant answers now!

y = x tan x , x = [ π , π ]


This graph has vertical asymptotes at x = ±π/2. The graph is symmetric about the y axis, so describing the left hand side will be sufficient. The function starts at (−π, 0) and decreases quickly to the asymptote. Then it starts on the other side of the asymptote in the second quadrant and decreases to the the origin.

Got questions? Get instant answers now!

y = x 2 sin ( x ) , x = [ −2 π , 2 π ]


This function starts at (−2π, 0), increases to near (−3π/2, 25), decreases through (−π, 0), achieves a local minimum and then increases through the origin. On the other side of the origin, the graph is the same but flipped, that is, it is congruent to the other half by a rotation of 180 degrees.

Got questions? Get instant answers now!

For f ( x ) = P ( x ) Q ( x ) to have an asymptote at y = 2 then the polynomials P ( x ) and Q ( x ) must have what relation?

Got questions? Get instant answers now!

For f ( x ) = P ( x ) Q ( x ) to have an asymptote at x = 0 , then the polynomials P ( x ) and Q ( x ) . must have what relation?

Q ( x ) . must have have x k + 1 as a factor, where P ( x ) has x k as a factor.

Got questions? Get instant answers now!

If f ( x ) has asymptotes at y = 3 and x = 1 , then f ( x ) has what asymptotes?

Got questions? Get instant answers now!

Both f ( x ) = 1 ( x 1 ) and g ( x ) = 1 ( x 1 ) 2 have asymptotes at x = 1 and y = 0 . What is the most obvious difference between these two functions?

lim x 1 f ( x ) and lim x 1 g ( x )

Got questions? Get instant answers now!

True or false: Every ratio of polynomials has vertical asymptotes.

Got questions? Get instant answers now!

Questions & Answers

I don't understand the formula
Adaeze Reply
who's formula
funny
What is a independent variable
Sifiso Reply
a variable that does not depend on another.
Andrew
solve number one step by step
bil Reply
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
Rochel Reply
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
Rochel Reply
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
Rochel Reply
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
not a bad question
Salim
what is function.
Nawaz Reply
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
vector
what is the power rule
Vanessa Reply
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
how do i deal with infinity in limits?
Itumeleng Reply
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
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