Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Limits

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #41 : Limits

Find the limit.

$\lim_{t \to 0}\frac{\sqrt{t^2+9}-3}{t^2}$

Possible Answers:

$\frac{1}{6}$

$\infty$

Correct answer:

$\frac{1}{6}$

Explanation:

By plugging in 0, we get

$\frac{\sqrt{0^2+9}-3}{0^2}=\frac{0}{0}$ .

This solution is "indeterminate." Instead of directly plugging in 0, we simplify the function first, so we multiply by the conjugate of the numerator:

$\frac{\sqrt{t^2+9}-3}{t^2}*\frac{\sqrt{t^2+9}+3}{\sqrt{t^2+9}+3}$ .

Multiplying the numerators (via the FOIL method) and the denominators, we get

$\\\frac{(\sqrt{t^2+9})^2+3\sqrt{t^2+9}-3\sqrt{t^2+9}-9}{t^2(\sqrt{t^2+9}+3)}\\\\=\frac{t^2+9-9}{t^2(\sqrt{t^2+9}+3)}\\\\=\frac{t^2}{t^2(\sqrt{t^2+9}+3)}\\\\=\frac{1}{\sqrt{t^2+9}+3}$

Plugging in 0, we get

$\frac{1}{\sqrt{0^2+9}+3}=\frac{1}{6}$

Report an Error

Example Question #42 : Limits

Find the value of the limit (if it exists):

$\lim_{x \to 2}\frac{x^2-2x}{x^2-x-2}$

Possible Answers:

Does not exist.

$\frac{2}{3}$

$\frac{-2}{3}$

Correct answer:

$\frac{2}{3}$

Explanation:

By "plugging in" 2, we get $\frac{0}{0}$ , which is indeterminate.

We need to manipulate the function to see if anything cancels out first.

By factoring, we have

$\lim_{x \to 2}\frac{x(x-2)}{(x+1)(x-2)}=\lim_{x \to 2}\frac{x}{(x+1)}=\frac{2}{3}$ .

Report an Error

Example Question #43 : Limits

Evaluate

$\lim_{x \to 0}\frac{\sqrt{x+4}-2}{x}$

Possible Answers:

$\frac{-1}{4}$

$\frac{1}{4}$

Correct answer:

$\frac{1}{4}$

Explanation:

Simplify the function first, then "plug in" 0 to evaluate the limit.

$\\\lim_{x \to 0}\frac{(\sqrt{x+4}-2)}{x}\frac{(\sqrt{x+4}+2)}{(\sqrt{x+4}+2)}\\\\=\lim_{x \to 0}\frac{(x+4)-4}{x(\sqrt{x+4}+2)}\\\\=\lim_{x \to 0}\frac{1}{\sqrt{x+4}+2}\\\\=\frac{1}{4}$

Report an Error

Example Question #1 : Finding Limits And One Sided Limits

Which graph is a possible sketch of the function that possesses the following characteristics:

$f(-2)= undefined; \lim_{x \rightarrow -2^-} f(x)= \infty; \lim_{x \rightarrow -2^+} f(x)= -\infty;$

Possible Answers:

Graph2

Graph3

There does not exist such a graph.

Graph4

Graph1

Correct answer:

Graph1

Explanation:

Since $\tiny { f(-2) = undefined}$ there is a possible vertical asymptote at x=-2 .

As we approach x=-2 from the left, the graph should tend to $\tiny { +\infty}$ . Approaching x=-2 from the right, the graph tends to $\tiny {-\infty}$ .

The only graph that does so is Graph1 .

Report an Error

Example Question #1 : Finding Limits And One Sided Limits

Rational_graph

The graph above is a sketch of the function . Find $\lim_{x \rightarrow 2^-}f(x)$ .

Possible Answers:

Does not exist

$-\infty$

$\infty$

Correct answer:

$-\infty$

Explanation:

We need to look at the behavior of the function as it tends to from the left. Therefore the answer is $\tiny { -\infty}$ .

Report an Error

Example Question #1 : Finding Limits And One Sided Limits

Rational_graph

The graph above is a sketch of . Find $\lim_{x \rightarrow -2} f(x)$ .

Possible Answers:

$-\infty$

Does not exist

$\infty$

Correct answer:

Does not exist

Explanation:

The limit does not exist because the one-sided limits are not equal; $\lim_{x \rightarrow -2^-}f(x) = -\infty$ , whereas $\lim_{x \rightarrow -2^+}f(x) = +\infty$ .

Report an Error

Example Question #3 : Finding Limits And One Sided Limits

Determine the value of $\lim_{x \to 5^-} \left |-5 \right |$ .

Possible Answers:

$-\infty$

$\infty$

Correct answer:

Explanation:

The expression ${}\lim_{x \to 5^-} \left |-5 \right |$ indicates that all points on the domain are equal to 5 since the absolute value negates negative values.

The $\lim_{x \to 5^-}$ is to search for the limit as the graph approaches x=5 from the left side of the graph. Since the absolute value of negative five is five, the graph approaches five from the left.

The correct answer is .

Report an Error

Example Question #2 : Finding Limits And One Sided Limits

Evaluate the following limit.

$\small \lim_{x\to 9} \frac{x-9}{\sqrt{x}-3}$

Possible Answers:

Limit does not exist

$\small 6$

$\small 3+\sqrt{3}$

Correct answer:

$\small 6$

Explanation:

This limit can be solved using simple manipulation of the expression inside the limit:

$\\ \small \small \lim_{x\to 9} \frac{x-9}{\sqrt{x}-3}\\ \\=\lim_{x\to 9} \frac{x-9}{\sqrt{x}-3}\cdot \frac{\sqrt{x}+3}{\sqrt{x}+3}\\ \\=\lim_{x\to 9} \frac{x-9}{x-9}\cdot (\sqrt{x}+3)\\ \\=\lim_{x\to 9} \sqrt{x}+3\\ \\=\sqrt{9}+3=6$

Report an Error

Example Question #3 : Finding Limits And One Sided Limits

Evaluatle the limit:

$\lim_{x \to 3} x^2-2x+6$

Possible Answers:

DNE

Correct answer:

Explanation:

Consider the domain of the function. Because this equation is a polynomial, x is not restricted by any value. Thus the way to evaluate this limit would simply be to plug the value that x is approaching into the limit equation.

$\lim_{x \to 3} x^2-2x+6=3^2-2*3+6=9-6+6=9$

Report an Error

Example Question #4 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to -2}\sqrt{x^4+3x+6}$

Possible Answers:

DNE

Correct answer:

Explanation:

$\lim_{x \to -2}\sqrt{x^4+3x+6}=\sqrt{(-2)^4+3(-2)+6}=\sqrt{16-6+6}=\sqrt{16}=4$

Report an Error

View Calculus 2 Tutors

Sharif
Certified Tutor

NCCU, Bachelors, Physics and Mathematics.

View Calculus 2 Tutors

Muhammad
Certified Tutor

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

View Calculus 2 Tutors

Alejandro
Certified Tutor

University of Western Ontario, Doctorate (PhD), Mathematics.

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Los Angeles, Math Tutors in Washington DC, Spanish Tutors in Boston, Spanish Tutors in Houston, Computer Science Tutors in Atlanta, GMAT Tutors in Boston, SSAT Tutors in New York City, Computer Science Tutors in San Diego, Chemistry Tutors in Denver, Spanish Tutors in Seattle

Popular Courses & Classes

LSAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Washington DC, LSAT Courses & Classes in Philadelphia, SAT Courses & Classes in Boston, ACT Courses & Classes in Chicago, SSAT Courses & Classes in Denver, SAT Courses & Classes in Atlanta, Spanish Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Washington DC, GMAT Courses & Classes in Atlanta

Popular Test Prep

SSAT Test Prep in Miami, ACT Test Prep in Washington DC, ISEE Test Prep in San Francisco-Bay Area, LSAT Test Prep in Boston, SAT Test Prep in Boston, ACT Test Prep in Boston, ISEE Test Prep in New York City, LSAT Test Prep in Seattle, SAT Test Prep in Atlanta, SSAT Test Prep in Houston

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

Calculus 2 : Limits

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #41 : Limits

Example Question #42 : Limits

Example Question #43 : Limits

Example Question #1 : Finding Limits And One Sided Limits

Example Question #1 : Finding Limits And One Sided Limits

Example Question #1 : Finding Limits And One Sided Limits

Example Question #3 : Finding Limits And One Sided Limits

Example Question #2 : Finding Limits And One Sided Limits

Example Question #3 : Finding Limits And One Sided Limits

Example Question #4 : Finding Limits And One Sided Limits

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: