4.6 Limits at infinity and asymptotes  (Page 9/14)

 Page 9 / 14

Consider the function $f\left(x\right)=5-{x}^{2\text{/}3}.$ Determine the point on the graph where a cusp is located. Determine the end behavior of $f.$

The function $f$ has a cusp at $\left(0,5\right)$ $\underset{x\to {0}^{-}}{\text{lim}}{f}^{\prime }\left(x\right)=\infty ,$ $\underset{x\to {0}^{+}}{\text{lim}}{f}^{\prime }\left(x\right)=\text{−}\infty .$ For end behavior, $\underset{x\to \text{±}\infty }{\text{lim}}f\left(x\right)=\text{−}\infty .$

Key concepts

• The limit of $f\left(x\right)$ is $L$ as $x\to \infty$ (or as $x\to \text{−}\infty \right)$ if the values $f\left(x\right)$ become arbitrarily close to $L$ as $x$ becomes sufficiently large.
• The limit of $f\left(x\right)$ is $\infty$ as $x\to \infty$ if $f\left(x\right)$ becomes arbitrarily large as $x$ becomes sufficiently large. The limit of $f\left(x\right)$ is $\text{−}\infty$ as $x\to \infty$ if $f\left(x\right)<0$ and $|f\left(x\right)|$ becomes arbitrarily large as $x$ becomes sufficiently large. We can define the limit of $f\left(x\right)$ as $x$ approaches $\text{−}\infty$ similarly.
• For a polynomial function $p\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\text{…}+{a}_{1}x+{a}_{0},$ where ${a}_{n}\ne 0,$ the end behavior is determined by the leading term ${a}_{n}{x}^{n}.$ If $n\ne 0,$ $p\left(x\right)$ approaches $\infty$ or $\text{−}\infty$ at each end.
• For a rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},$ 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 $y=\frac{{a}_{n}}{{b}_{n}}$ 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 $\infty$ or $\text{−}\infty$ at each end.

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

$x=1$

$x=-1,x=2$

$x=0$

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

$f\left(x\right)=\frac{x+1}{{x}^{2}+5x+4},a=-1$

$f\left(x\right)=\frac{x}{x-2},a=2$

Yes, there is a vertical asymptote

$f\left(x\right)={\left(x+2\right)}^{3\text{/}2},a=-2$

$f\left(x\right)={\left(x-1\right)}^{-1\text{/}3},a=1$

Yes, there is vertical asymptote

$f\left(x\right)=1+{x}^{-2\text{/}5},a=1$

For the following exercises, evaluate the limit.

$\underset{x\to \infty }{\text{lim}}\frac{1}{3x+6}$

$0$

$\underset{x\to \infty }{\text{lim}}\frac{2x-5}{4x}$

$\underset{x\to \infty }{\text{lim}}\frac{{x}^{2}-2x+5}{x+2}$

$\infty$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{3{x}^{3}-2x}{{x}^{2}+2x+8}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{{x}^{4}-4{x}^{3}+1}{2-2{x}^{2}-7{x}^{4}}$

$-\frac{1}{7}$

$\underset{x\to \infty }{\text{lim}}\frac{3x}{\sqrt{{x}^{2}+1}}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{\sqrt{4{x}^{2}-1}}{x+2}$

$-2$

$\underset{x\to \infty }{\text{lim}}\frac{4x}{\sqrt{{x}^{2}-1}}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{4x}{\sqrt{{x}^{2}-1}}$

$-4$

$\underset{x\to \infty }{\text{lim}}\frac{2\sqrt{x}}{x-\sqrt{x}+1}$

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

$f\left(x\right)=x-\frac{9}{x}$

Horizontal: none, vertical: $x=0$

$f\left(x\right)=\frac{1}{1-{x}^{2}}$

$f\left(x\right)=\frac{{x}^{3}}{4-{x}^{2}}$

Horizontal: none, vertical: $x=\text{±}2$

$f\left(x\right)=\frac{{x}^{2}+3}{{x}^{2}+1}$

$f\left(x\right)=\text{sin}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\left(2x\right)$

Horizontal: none, vertical: none

$f\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x+\text{cos}\left(3x\right)+\text{cos}\left(5x\right)$

$f\left(x\right)=\frac{x\phantom{\rule{0.1em}{0ex}}\text{sin}\left(x\right)}{{x}^{2}-1}$

Horizontal: $y=0,$ vertical: $x=\text{±}1$

$f\left(x\right)=\frac{x}{\text{sin}\left(x\right)}$

$f\left(x\right)=\frac{1}{{x}^{3}+{x}^{2}}$

Horizontal: $y=0,$ vertical: $x=0$ and $x=-1$

$f\left(x\right)=\frac{1}{x-1}-2x$

$f\left(x\right)=\frac{{x}^{3}+1}{{x}^{3}-1}$

Horizontal: $y=1,$ vertical: $x=1$

$f\left(x\right)=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x+\text{cos}\phantom{\rule{0.1em}{0ex}}x}{\text{sin}\phantom{\rule{0.1em}{0ex}}x-\text{cos}\phantom{\rule{0.1em}{0ex}}x}$

$f\left(x\right)=x-\text{sin}\phantom{\rule{0.1em}{0ex}}x$

Horizontal: none, vertical: none

$f\left(x\right)=\frac{1}{x}-\sqrt{x}$

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

$x=1$ and $y=2$

Answers will vary, for example: $y=\frac{2x}{x-1}$

$x=1$ and $y=0$

$y=4,$ $x=-1$

Answers will vary, for example: $y=\frac{4x}{x+1}$

$x=0$

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

[T] $f\left(x\right)=\frac{1}{x+10}$

$y=0$

[T] $f\left(x\right)=\frac{x+1}{{x}^{2}+7x+6}$

[T] $\underset{x\to \text{−}\infty }{\text{lim}}{x}^{2}+10x+25$

$\infty$

[T] $\underset{x\to \text{−}\infty }{\text{lim}}\frac{x+2}{{x}^{2}+7x+6}$

[T] $\underset{x\to \infty }{\text{lim}}\frac{3x+2}{x+5}$

$y=3$

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.

$y=3{x}^{2}+2x+4$

$y={x}^{3}-3{x}^{2}+4$

$y=\frac{2x+1}{{x}^{2}+6x+5}$

$y=\frac{{x}^{3}+4{x}^{2}+3x}{3x+9}$

$y=\frac{{x}^{2}+x-2}{{x}^{2}-3x-4}$

$y=\sqrt{{x}^{2}-5x+4}$

$y=2x\sqrt{16-{x}^{2}}$

$y=\frac{\text{cos}\phantom{\rule{0.1em}{0ex}}x}{x},$ on $x=\left[-2\pi ,2\pi \right]$

$y={e}^{x}-{x}^{3}$

$y=x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}x,x=\left[\text{−}\pi ,\pi \right]$

$y=x\phantom{\rule{0.1em}{0ex}}\text{ln}\left(x\right),x>0$

$y={x}^{2}\text{sin}\left(x\right),x=\left[-2\pi ,2\pi \right]$

For $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ to have an asymptote at $y=2$ then the polynomials $P\left(x\right)$ and $Q\left(x\right)$ must have what relation?

For $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ to have an asymptote at $x=0,$ then the polynomials $P\left(x\right)$ and $Q\left(x\right).$ must have what relation?

$Q\left(x\right).$ must have have ${x}^{k+1}$ as a factor, where $P\left(x\right)$ has ${x}^{k}$ as a factor.

If ${f}^{\prime }\left(x\right)$ has asymptotes at $y=3$ and $x=1,$ then $f\left(x\right)$ has what asymptotes?

Both $f\left(x\right)=\frac{1}{\left(x-1\right)}$ and $g\left(x\right)=\frac{1}{{\left(x-1\right)}^{2}}$ have asymptotes at $x=1$ and $y=0.$ What is the most obvious difference between these two functions?

$\underset{x\to {1}^{-}}{\text{lim}}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\underset{x\to {1}^{-}}{\text{lim}}g\left(x\right)$

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

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

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