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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 ) = .

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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

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

Yes, there is a vertical asymptote

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f ( x ) = ( x + 2 ) 3 / 2 , a = −2

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f ( x ) = ( x 1 ) −1 / 3 , a = 1

Yes, there is vertical asymptote

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f ( x ) = 1 + x −2 / 5 , a = 1

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For the following exercises, evaluate the limit.

lim x 1 3 x + 6

0

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lim x 2 x 5 4 x

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lim x x 2 2 x + 5 x + 2

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

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

1 7

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lim x 3 x x 2 + 1

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lim x 4 x 2 1 x + 2

−2

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lim x 4 x x 2 1

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lim x 4 x x 2 1

−4

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lim x 2 x x x + 1

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

f ( x ) = x 9 x

Horizontal: none, vertical: x = 0

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f ( x ) = x 3 4 x 2

Horizontal: none, vertical: x = ± 2

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

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

Horizontal: none, vertical: none

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

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

Horizontal: y = 0 , vertical: x = ± 1

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f ( x ) = 1 x 3 + x 2

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

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f ( x ) = 1 x 1 2 x

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

Horizontal: y = 1 , vertical: x = 1

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

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f ( x ) = x sin x

Horizontal: none, vertical: none

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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

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y = 4 , x = −1

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

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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

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

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

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

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

y = 3

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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

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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.

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

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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).

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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.

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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.

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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?

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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.

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

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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 )

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

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Questions & Answers

find the domain and range of f(x)= 4x-7/x²-6x+8
Nick Reply
find the range of f(x)=(x+1)(x+4)
Jane Reply
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
Harrieta Reply
d/dx{1/y - lny + X^3.Y^5}
mogomotsi Reply
How to identify domain and range
Umar Reply
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
Adri
sorry
Dr
hi adri ana
Dr
:(
Shun
was up
Dr
hello
Adarsh
is it chatting app?.. I do not see any calculus here. lol
Adarsh
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
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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