We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 … 12 13 14 15 16 17 18 19 20 21 Next →

Example Question #17 : Calculating Limits Using Algebra

$\begin{align*}&\text{Determine the limit if it exists of }\frac{73x + 3x^{2} + 342}{52x + 4x^{2} - 360}\\&\text{At }x=-18\end{align*}$

Possible Answers:

$-\frac{92}{35}$

$\frac{35}{92}$

$-\frac{35}{92}$

$\frac{92}{35}$

Correct answer:

$\frac{35}{92}$

Explanation:

$\begin{align*}&\text{Simply plugging in the value of }x=-18\text{ into the function}\\&\frac{73x + 3x^{2} + 342}{52x + 4x^{2} - 360}\\&\text{will not tell us much, because it simply gives us a zero in}\\&\text{the numerator and denominator: }\frac{0}{0}\\&\text{Note, however, that we're calculating a limit. As such, we can try to simplify our fraction.}\\&\text{We can do this because we're not actually evaluating the function at these zero points,}\\&\text{just close to them. Factoring our function we find:}\\&\frac{(3x + 19)\cdot (x + 18)}{4\cdot (x - 5)\cdot (x + 18)}\\&\text{Which reduces to:}\\&\frac{(3x + 19)}{(4x - 20)}\\&\text{At the value of }x=-18\text{ the function is}\\&\frac{35}{92}\\&\text{*Note*: To find roots, you can always utilize the quadratic formula:}\\&\text{For equations of the form: }ax^2+bx+c\\&\text{The roots are: }x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{align*}$

Report an Error

Example Question #18 : Calculating Limits Using Algebra

$\begin{align*}&\text{Evaluate the limit of }\frac{3x^{2} - 24x + 36}{x - 6}\\&\text{At the point }x=6\end{align*}$

Possible Answers:

$\frac{1}{12}$

$-\frac{1}{12}$

Correct answer:

Explanation:

$\begin{align*}&\text{Simply plugging in the value of }x=6\text{ into the function}\\&\frac{3x^{2} - 24x + 36}{x - 6}\\&\text{will not tell us much, because it simply gives us a zero in}\\&\text{the numerator and denominator: }\frac{0}{0}\\&\text{Note, however, that we're calculating a limit. As such, we can try to simplify our fraction.}\\&\text{We can do this because we're not actually evaluating the function at these zero points,}\\&\text{just close to them. Factoring our function we find:}\\&\frac{3\cdot (x - 2)\cdot (x - 6)}{x - 6}\\&\text{Which reduces to:}\\&3x - 6\\&\text{At the value of }x=6\text{ the function is}\\&12\\&\text{*Note*: To find roots, you can always utilize the quadratic formula:}\\&\text{For equations of the form: }ax^2+bx+c\\&\text{The roots are: }x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{align*}$

Report an Error

Example Question #11 : Calculating Limits Using Algebra

$\begin{align*}&\text{Evaluate the limit of }\frac{2x^{2} - 38x + 68}{5x^{2} - 110x + 425}\\&\text{At the point }x=17\end{align*}$

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

$\begin{align*}&\text{Simply plugging in the value of }x=17\text{ into the function}\\&\frac{2x^{2} - 38x + 68}{5x^{2} - 110x + 425}\\&\text{will not tell us much, because it simply gives us a zero in}\\&\text{the numerator and denominator: }\frac{0}{0}\\&\text{Note, however, that we're calculating a limit. As such, we can try to simplify our fraction.}\\&\text{We can do this because we're not actually evaluating the function at these zero points,}\\&\text{just close to them. Factoring our function we find:}\\&\frac{2\cdot (x - 2)\cdot (x - 17)}{5\cdot (x - 5)\cdot (x - 17)}\\&\text{Which reduces to:}\\&\frac{(2x - 4)}{(5x - 25)}\\&\text{At the value of }x=17\text{ the function is}\\&\frac{1}{2}\\&\text{*Note*: To find roots, you can always utilize the quadratic formula:}\\&\text{For equations of the form: }ax^2+bx+c\\&\text{The roots are: }x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{align*}$

Report an Error

Example Question #20 : Calculating Limits Using Algebra

$\begin{align*}&\text{Find the limit of }f(x)=\frac{4x^{2} - 68x - 152}{2x^{2} - 62x + 456}\\&\text{At the location }x=19\end{align*}$

Possible Answers:

$-\frac{1}{6}$

$\frac{1}{6}$

Correct answer:

Explanation:

$\begin{align*}&\text{Simply plugging in the value of }x=19\text{ into the function}\\&\frac{4x^{2} - 68x - 152}{2x^{2} - 62x + 456}\\&\text{will not tell us much, because it simply gives us a zero in}\\&\text{the numerator and denominator: }\frac{0}{0}\\&\text{Note, however, that we're calculating a limit. As such, we can try to simplify our fraction.}\\&\text{We can do this because we're not actually evaluating the function at these zero points,}\\&\text{just close to them. Factoring our function we find:}\\&\frac{4\cdot (x + 2)\cdot (x - 19)}{2\cdot (x - 12)\cdot (x - 19)}\\&\text{Which reduces to:}\\&\frac{(4x + 8)}{(2x - 24)}\\&\text{At the value of }x=19\text{ the function is}\\&6\\&\text{*Note*: To find roots, you can always utilize the quadratic formula:}\\&\text{For equations of the form: }ax^2+bx+c\\&\text{The roots are: }x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{align*}$

Report an Error

Example Question #21 : Limits Of Functions (Including One Sided Limits)

$\begin{align*}&\text{Evaluate the limit of }\frac{85x + 5x^{2} - 90}{76x + 5x^{2} - 252}\\&\text{At the point }x=-18\end{align*}$

Possible Answers:

$\frac{95}{104}$

$\frac{104}{95}$

$-\frac{95}{104}$

$-\frac{104}{95}$

Correct answer:

$\frac{95}{104}$

Explanation:

$\begin{align*}&\text{Simply plugging in the value of }x=-18\text{ into the function}\\&\frac{85x + 5x^{2} - 90}{76x + 5x^{2} - 252}\\&\text{will not tell us much, because it simply gives us a zero in}\\&\text{the numerator and denominator: }\frac{0}{0}\\&\text{Note, however, that we're calculating a limit. As such, we can try to simplify our fraction.}\\&\text{We can do this because we're not actually evaluating the function at these zero points,}\\&\text{just close to them. Factoring our function we find:}\\&\frac{5\cdot (x - 1)\cdot (x + 18)}{(5x - 14)\cdot (x + 18)}\\&\text{Which reduces to:}\\&\frac{(5x - 5)}{(5x - 14)}\\&\text{At the value of }x=-18\text{ the function is}\\&\frac{95}{104}\\&\text{*Note*: To find roots, you can always utilize the quadratic formula:}\\&\text{For equations of the form: }ax^2+bx+c\\&\text{The roots are: }x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{align*}$

Report an Error

Example Question #21 : Limits Of Functions (Including One Sided Limits)

Find the limit of the following function as x approaches infinity.

$\frac{1}{x}$

Possible Answers:

$\frac{1}{4}$

Infinity

Correct answer:

Explanation:

As x becomes infinitely large, $\frac{1}{x}$ approaches .

Report an Error

Example Question #21 : Calculating Limits Using Algebra

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \\ 0 &x= 0 \\ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Possible Answers:

$\lim_{x\rightarrow 0 }f(x)$ does not exist, because either or both of $\lim_{x\rightarrow 0 - }f(x)$ and $\lim_{x\rightarrow 0 + }f(x)$ is unequal to f(0) .

$\lim_{x\rightarrow 0 }f(x)$ does not exist, because $\lim_{x\rightarrow 0 - }f(x) \ne \lim_{x\rightarrow 0 + }f(x)$ .

$\frac{1}{2}$

Correct answer:

Explanation:

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

Report an Error

Example Question #21 : Calculating Limits Using Algebra

Define $f (x) = \begin{Bmatrix} x^{2}+2x+ 3& x< 0 \\ a \cos\theta x & x \ge 0 \end{matrix}$ for some real $a,\theta$ .

Evaluate and $\theta$ so that is both continuous and differentiable at x = 0 .

Possible Answers:

$a = 3, \theta = 2$

No such values exist.

$a = 2, \theta = 3$

$a = 3, \theta = 1$

$a = 1, \theta = 3$

Correct answer:

No such values exist.

Explanation:

For to be continuous, it must hold that

$\lim_{x \rightarrow 0^{-}} f(x) = \lim_{x \rightarrow 0^{+}} f(x)$ .

To find $\lim_{x \rightarrow 0^{-}} f(x)$ , we can use the definition of f(x) for all negative values of :

$\lim_{x \rightarrow 0^{-}} f(x) =\lim_{x \rightarrow 0^{-}} ( x^{2}+2x+ 3) = 0^{2}+ 2(0)+ 3 = 3$

It must hold that $\lim_{x \rightarrow 0^{+}} f(x) =3$ as well; using the definition of f(x) for all positive values of :

$\lim_{x \rightarrow 0^{+}} f(x) =\lim_{x \rightarrow 0^{+}} a \cos \theta x = a \cos 0 = a \cdot 1 = a$

$\lim_{x \rightarrow 0^{+}} f(x) =3$ , so a = 3 .

Now examine f'(x) . For to be differentiable, it must hold that

$\lim_{x \rightarrow 0^{-}} f'(x) = \lim_{x \rightarrow 0^{+}} f'(x)$ .

To find $\lim_{x \rightarrow 0^{-}} f'(x)$ , we can differentiate the expression for f(x) for all negative values of :

$f(x)= x^{2}+ 2x + 3$

f'(x) = 2x + 2

$\lim_{x \rightarrow 0^{-}} f'(x) = \lim_{x \rightarrow 0^{-}} (2x +2)= 2(0)+2 = 2$

To find $\lim_{x \rightarrow 0^{+}} f'(x)$ , we can differentiate the expression for f(x) for all positive values of :

$f(x) = a \cos \theta x$

$f'(x) = - a \theta \sin \theta x$

$\lim_{x \rightarrow 0^{+}} f'(x) =\lim_{x \rightarrow 0^{+}} ( - a \theta \sin \theta x)$

We know that a = 3 , so

$\lim_{x \rightarrow 0^{+}} f'(x) =\lim_{x \rightarrow 0^{+}} ( - 3 \theta \sin \theta x) = -3 \theta \sin 0 = -3 \theta \cdot 0 = 0$

Since $\lim_{x \rightarrow 0^{-}} f'(x) = 2$ and $\lim_{x \rightarrow 0^{+}} f'(x)=0$ ,

$\lim_{x \rightarrow 0 } f'(x)$ cannot exist regardless of the values of and $\theta$ .

Report an Error

Example Question #22 : Limits Of Functions (Including One Sided Limits)

Find the following limit as x approaches infinity.

$\ln (x)$

Possible Answers:

$\infty$

$-\infty$

Correct answer:

$\infty$

Explanation:

As x becomes infinitely large, $\ln (x)$ approaches infinity.

Report an Error

Example Question #26 : Limits Of Functions (Including One Sided Limits)

Find the limit:

$\lim_{x\rightarrow \propto }\frac{4x^3+3x^2+10}{10x^3+x^2+8x}$

Possible Answers:

$\frac{4}{10}$

$\propto$

DNE (Does not exist)

Correct answer:

$\frac{4}{10}$

Explanation:

So for limits involving infinity, there is one important concept regarding fractions that is important to understand.

So if you have a fraction with a number on the bottom that is getting larger and larger, the whole fraction becomes smaller.

For example $\frac{1}{1000}>\frac{1}{10000}$

So the way to solve for limits involving infinity is to divide each term on the top, and each term on the bottom by the largest power of the variable. In the case of this problem, that is x^3 .

So if you do this you get:

$\lim_{x\rightarrow \propto }\frac{\frac{4x^3}{x^3}+\frac{3x^2}{x^3}+\frac{10}{x^3}}{\frac{10x^3}{x^3}+\frac{x^2}{x^3}+\frac{8x}{x^3}}$

So anything divided by itself is 1, which means that the first two terms cancel to one. Then the rest of the terms with a higher power of x on the bottom than on the top will have some power of x left on the bottom. This will look like:

$\lim_{x\rightarrow \propto }\frac{4+\frac{3}{x}+\frac{10}{x^3}}{10+\frac{1}{x}+\frac{8}{x^2}}$

Then, if you let the limit go to infinity, the terms with an x left on the bottom will go to zero.

This leaves you with the answer of $\frac{4}{10}$

Report an Error

← Previous 1 2 … 12 13 14 15 16 17 18 19 20 21 Next →

View AP Calculus AB Tutors

Yucheng
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Mechanical Engineering.

View AP Calculus AB Tutors

Zhi Xuan
Certified Tutor

Queen's University, Bachelor, Engineering .

View AP Calculus AB Tutors

Michael
Certified Tutor

University of Richmond, Bachelor of Science, Mathematics. Central Connecticut State University, Master of Arts, Mathematics.

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Spanish Tutors in Houston, Spanish Tutors in Phoenix, Physics Tutors in Phoenix, Spanish Tutors in New York City, Spanish Tutors in Seattle, ISEE Tutors in Boston, SSAT Tutors in Dallas Fort Worth, SSAT Tutors in Denver, Computer Science Tutors in Dallas Fort Worth, Reading Tutors in San Diego

Popular Courses & Classes

ACT Courses & Classes in Chicago, Spanish Courses & Classes in Denver, SSAT Courses & Classes in Chicago, SAT Courses & Classes in Denver, GRE Courses & Classes in San Diego, Spanish Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Philadelphia, SSAT Courses & Classes in New York City, GRE Courses & Classes in Phoenix, Spanish Courses & Classes in San Diego

Popular Test Prep

GRE Test Prep in Phoenix, SAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Dallas Fort Worth, ACT Test Prep in Los Angeles, ISEE Test Prep in Denver, GMAT Test Prep in San Diego, SAT Test Prep in Philadelphia, GMAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Miami, ISEE Test Prep in New York City

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #17 : Calculating Limits Using Algebra

Example Question #18 : Calculating Limits Using Algebra

Example Question #11 : Calculating Limits Using Algebra

Example Question #20 : Calculating Limits Using Algebra

Example Question #21 : Limits Of Functions (Including One Sided Limits)

Example Question #21 : Limits Of Functions (Including One Sided Limits)

Example Question #21 : Calculating Limits Using Algebra

Example Question #21 : Calculating Limits Using Algebra

Example Question #22 : Limits Of Functions (Including One Sided Limits)

Example Question #26 : Limits Of Functions (Including One Sided Limits)

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: