# 12.2 Finding limits: properties of limits  (Page 4/5)

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Access the following online resource for additional instruction and practice with properties of limits.

## Key concepts

• The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See [link] .
• The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See [link] and [link] .
• The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See [link] .
• The limit of the root of a function equals the corresponding root of the limit of the function.
• One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See [link] .
• Another method of finding the limit of a complex fraction is to find the LCD. See [link] .
• A limit containing a function containing a root may be evaluated using a conjugate. See [link] .
• The limits of some functions expressed as quotients can be found by factoring. See [link] .
• One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See [link] .

## Verbal

Give an example of a type of function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ whose limit, as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a,$ is $\text{\hspace{0.17em}}f\left(a\right).$

If $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is a polynomial function, the limit of a polynomial function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ will always be $\text{\hspace{0.17em}}f\left(a\right).$

When direct substitution is used to evaluate the limit of a rational function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and the result is $\text{\hspace{0.17em}}f\left(a\right)=\frac{0}{0},$ does this mean that the limit of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ does not exist?

What does it mean to say the limit of $\text{\hspace{0.17em}}f\left(x\right),$ as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}c,$ is undefined?

It could mean either (1) the values of the function increase or decrease without bound as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}c,$ or (2) the left and right-hand limits are not equal.

## Algebraic

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 0}{\mathrm{lim}}\left(3\right)$

$\underset{x\to 2}{\mathrm{lim}}\left(\frac{-5x}{{x}^{2}-1}\right)$

$\frac{-10}{3}$

$\underset{x\to 2}{\mathrm{lim}}\left(\frac{{x}^{2}-5x+6}{x+2}\right)$

$\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}-9}{x-3}\right)$

6

$\underset{x\to -1}{\mathrm{lim}}\left(\frac{{x}^{2}-2x-3}{x+1}\right)$

$\underset{x\to \frac{3}{2}}{\mathrm{lim}}\left(\frac{6{x}^{2}-17x+12}{2x-3}\right)$

$\frac{1}{2}$

$\underset{x\to -\frac{7}{2}}{\mathrm{lim}}\left(\frac{8{x}^{2}+18x-35}{2x+7}\right)$

$\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}-9}{x-5x+6}\right)$

6

$\underset{x\to -3}{\mathrm{lim}}\left(\frac{-7{x}^{4}-21{x}^{3}}{-12{x}^{4}+108{x}^{2}}\right)$

$\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}+2x-3}{x-3}\right)$

does not exist

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(3+h\right)}^{3}-27}{h}\right)$

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(2-h\right)}^{3}-8}{h}\right)$

$-12$

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(h+3\right)}^{2}-9}{h}\right)$

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{\sqrt{5-h}-\sqrt{5}}{h}\right)$

$-\frac{\sqrt{5}}{10}$

$\underset{x\to 0}{\mathrm{lim}}\left(\frac{\sqrt{3-x}-\sqrt{3}}{x}\right)$

$\underset{x\to 9}{\mathrm{lim}}\left(\frac{{x}^{2}-81}{3-\sqrt{x}}\right)$

$-108$

$\underset{x\to 1}{\mathrm{lim}}\left(\frac{\sqrt{x}-{x}^{2}}{1-\sqrt{x}}\right)$

$\underset{x\to 0}{\mathrm{lim}}\left(\frac{x}{\sqrt{1+2x}-1}\right)$

1

$\underset{x\to \frac{1}{2}}{\mathrm{lim}}\left(\frac{{x}^{2}-\frac{1}{4}}{2x-1}\right)$

$\underset{x\to 4}{\mathrm{lim}}\left(\frac{{x}^{3}-64}{{x}^{2}-16}\right)$

6

$\underset{x\to {2}^{-}}{\mathrm{lim}}\left(\frac{|x-2|}{x-2}\right)$

$\underset{x\to {2}^{+}}{\mathrm{lim}}\left(\frac{|x-2|}{x-2}\right)$

1

$\underset{x\to 2}{\mathrm{lim}}\left(\frac{|x-2|}{x-2}\right)$

$\underset{x\to {4}^{-}}{\mathrm{lim}}\left(\frac{|x-4|}{4-x}\right)$

1

$\underset{x\to {4}^{+}}{\mathrm{lim}}\left(\frac{|x-4|}{4-x}\right)$

$\underset{x\to 4}{\mathrm{lim}}\left(\frac{|x-4|}{4-x}\right)$

does not exist

$\underset{x\to 2}{\mathrm{lim}}\left(\frac{-8+6x-{x}^{2}}{x-2}\right)$

For the following exercise, use the given information to evaluate the limits: $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}f\left(x\right)=3,$ $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}g\left(x\right)=5$

$\underset{x\to c}{\mathrm{lim}}\text{\hspace{0.17em}}\left[\text{\hspace{0.17em}}2f\left(x\right)+\sqrt{g\left(x\right)}\text{\hspace{0.17em}}\right]$

$6+\sqrt{5}$

$\underset{x\to c}{\mathrm{lim}}\text{\hspace{0.17em}}\left[\text{\hspace{0.17em}}3f\left(x\right)+\sqrt{g\left(x\right)}\text{\hspace{0.17em}}\right]$

$\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}$

$\frac{3}{5}$

For the following exercises, evaluate the following limits.

$\underset{x\to 2}{\mathrm{lim}}\mathrm{cos}\left(\pi x\right)$

$\underset{x\to 2}{\mathrm{lim}}\mathrm{sin}\left(\pi x\right)$

0

$\underset{x\to 2}{\mathrm{lim}}\mathrm{sin}\left(\frac{\pi }{x}\right)$

$-3$

does not exist; right-hand limit is not the same as the left-hand limit.

$\underset{x\to 4}{\mathrm{lim}}\frac{\sqrt{x+5}-3}{x-4}$

$\underset{x\to {2}^{+}}{\mathrm{lim}}\left(2x-\mathrm{〚x〛}\right)$

2

$\underset{x\to 2}{\mathrm{lim}}\frac{\sqrt{x+7}-3}{{x}^{2}-x-2}$

$\underset{x\to {3}^{+}}{\mathrm{lim}}\frac{{x}^{2}}{{x}^{2}-9}$

Limit does not exist; limit approaches infinity.

For the following exercises, find the average rate of change $\text{\hspace{0.17em}}\frac{f\left(x+h\right)-f\left(x\right)}{h}.$

$f\left(x\right)=x+1$

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

$4x+2h$

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

$f\left(x\right)={x}^{2}+4x-100$

$2x+h+4$

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

$f\left(x\right)=\mathrm{cos}\left(x\right)$

$\frac{\mathrm{cos}\left(x+h\right)-\mathrm{cos}\left(x\right)}{h}$

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

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

$\frac{-1}{x\left(x+h\right)}$

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

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

$\frac{-1}{\sqrt{x+h}+\sqrt{x}}$

## Graphical

Find an equation that could be represented by [link] .

Find an equation that could be represented by [link] .

$f\left(x\right)=\frac{{x}^{2}+5x+6}{x+3}$

For the following exercises, refer to [link] .

What is the right-hand limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 0?

What is the left-hand limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 0?

does not exist

## Real-world applications

The position function $\text{\hspace{0.17em}}s\left(t\right)=-16{t}^{2}+144t\text{\hspace{0.17em}}$ gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval $\text{\hspace{0.17em}}\left[1,2\right]$ .

The height of a projectile is given by $\text{\hspace{0.17em}}s\left(t\right)=-64{t}^{2}+192t\text{\hspace{0.17em}}$ Find the average rate of change of the height from $\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ second to $\text{\hspace{0.17em}}t=1.5\text{\hspace{0.17em}}$ seconds.

52

The amount of money in an account after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years compounded continuously at 4.25% interest is given by the formula $\text{\hspace{0.17em}}A={A}_{0}{e}^{0.0425t},$ where $\text{\hspace{0.17em}}{A}_{0}\text{\hspace{0.17em}}$ is the initial amount invested. Find the average rate of change of the balance of the account from $\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ year to $\text{\hspace{0.17em}}t=2\text{\hspace{0.17em}}$ years if the initial amount invested is $1,000.00. #### Questions & Answers I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once Carlos Reply How can you tell what type of parent function a graph is ? Mary Reply generally by how the graph looks and understanding what the base parent functions look like and perform on a graph William if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero William y=x will obviously be a straight line with a zero slope William y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis William y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer. Aaron yes, correction on my end, I meant slope of 1 instead of slope of 0 William what is f(x)= Karim Reply I don't understand Joe Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain." Thomas Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-) Thomas GREAT ANSWER THOUGH!!! Darius Thanks. Thomas Â Thomas It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â Thomas Now it shows, go figure? Thomas what is this? unknown Reply i do not understand anything unknown lol...it gets better Darius I've been struggling so much through all of this. my final is in four weeks 😭 Tiffany this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts Darius thank you I have heard of him. I should check him out. Tiffany is there any question in particular? Joe I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously. Tiffany Sure, are you in high school or college? Darius Hi, apologies for the delayed response. I'm in college. Tiffany how to solve polynomial using a calculator Ef Reply So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right? KARMEL Reply The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26 Rima Reply The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer? Rima I done know Joe What kind of answer is that😑? Rima I had just woken up when i got this message Joe Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that Rima i have a question. Abdul how do you find the real and complex roots of a polynomial? Abdul @abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up Nare This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1 Abdul @Nare please let me know if you can solve it. Abdul I have a question juweeriya hello guys I'm new here? will you happy with me mustapha The average annual population increase of a pack of wolves is 25. Brittany Reply how do you find the period of a sine graph Imani Reply Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period Am if not then how would I find it from a graph Imani by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates. Am you could also do it with two consecutive minimum points or x-intercepts Am I will try that thank u Imani Case of Equilateral Hyperbola Jhon Reply ok Zander ok Shella f(x)=4x+2, find f(3) Benetta f(3)=4(3)+2 f(3)=14 lamoussa 14 Vedant pre calc teacher: "Plug in Plug in...smell's good" f(x)=14 Devante 8x=40 Chris Explain why log a x is not defined for a < 0 Baptiste Reply the sum of any two linear polynomial is what Esther Reply divide simplify each answer 3/2÷5/4 Momo Reply divide simplify each answer 25/3÷5/12 Momo how can are find the domain and range of a relations austin Reply the range is twice of the natural number which is the domain Morolake A cell phone company offers two plans for minutes. Plan A:$15 per month and $2 for every 300 texts. Plan B:$25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money? Diddy Reply 6000 Robert more than 6000 Robert For Plan A to reach$27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert