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 #1 : Limits

Evaluate:

$\lim_{x\rightarrow 1} \frac{8^{x}- 8}{2^{x}- 2}$

Possible Answers:

$\infty$

The limit does not exist.

None of the other answers is correct.

$\ln 4$

Correct answer:

None of the other answers is correct.

Explanation:

Evaluated at x = 1 , the numerator and the denominator are both equal to 0, as shown below:

$8^{x}- 8 = 8^{1}- 8 = 8 - 8 = 0$

$2^{x}- 2 = 2^{1}- 2 = 2 - 2 = 0$

So a straightforward substitution will not work. L'Hospital's rule will work here, but an easier way is to note that

$8 = 2 ^{3}$ and $8^{x} = \left ( 2 ^{3} \right )^{x} = 2 ^{3x} = \left ( 2 ^{x} \right )^{3}$ .

So the expression can be rewritten - and solved - as follows:

$\lim_{x\rightarrow 1} \frac{8^{x}- 8}{2^{x}- 2}$

$= \lim_{x\rightarrow 1} \frac{\left (2^{x} \right )^{3}- 2^{3}}{2^{x}- 2}$

$= \lim_{x\rightarrow 1} \frac{\left (2^{x}- 2 \right) \left [ \left ( 2^{x} \right ) ^{2} +2 \cdot 2^{x} + 2 ^{2} \right ] }{2^{x}- 2}$

$= \lim_{x\rightarrow 1} \left [ \left ( 2^{x} \right ) ^{2} +2 \cdot 2^{x} + 2 ^{2} \right ]$

$= \left [ \left ( 2^{1} \right ) ^{2} +2 \cdot 2^{1} + 2 ^{2} \right ] = 2^{2} + 2 \cdot 2 + 2^{2} = 4 + 4 + 4 = 12$

Report an Error

Example Question #1 : Limit Concepts

Evaluate the limit:

$\lim_{y \rightarrow 0} \frac{\tan2y}{\sin 5y}$

Possible Answers:

Does Not Exist

$\frac{2}{5}$

$\sin \frac{2}{5}$

$\frac{5}{2}$

Correct answer:

$\frac{2}{5}$

Explanation:

Directly evaluating the limit will produce an indeterminant answer of $\frac{0}{0}$ .

Rewriting the limit in terms of sine and cosine, $\lim_{y \rightarrow 0} \frac{\sin 2y}{\cos 2y} \cdot \frac{1}{\sin 5y}$ , we can try to manipulate the function in order to utilize the property $\lim_{x \rightarrow 0} \frac{\sin x}{x}=1$ .

Multiplying the function by the arguments of the sine functions, $\lim_{y \rightarrow 0} \frac{\sin 2y}{2y} \cdot \frac{1}{\cos 2y} \cdot \frac{5y}{\sin 5y} \cdot \frac{2y}{5y}$ , we can see that the limit will be $(1)(1)(1)\left(\frac{2}{5}\right) = \left(\frac{2}{5} \right )$ .

Report an Error

Example Question #2 : Limits

Evaluate $\lim_{x\rightarrow \frac{\pi}{2}^{-} } \cos x \tan x$ .

Possible Answers:

$- \infty$

$\infty$

The limit does not exist.

Correct answer:

Explanation:

$\lim_{x\rightarrow \frac{\pi}{2}^{-} } \cos x = \cos \frac{\pi}{2} = 0$

and

$\lim_{x\rightarrow \frac{\pi}{2}^{-} } \tan x = \infty$ ,

so we cannot solve this by substituting.

However, we can rewrite the expression:

$\lim_{x\rightarrow \frac{\pi}{2}^{-} } \cos x \cdot \frac{\sin x}{\cos x}$

$= \lim_{x\rightarrow \frac{\pi}{2}^{-} } \sin x = \sin \frac{\pi}{2} = 1$

Report an Error

Example Question #2 : Limit Concepts

Find the limit of $\frac{sin^5(x)}{x^5}$ as approaches infinity.

Possible Answers:

$\infty$

Inconclusive

$\frac{1}{2}$

Correct answer:

Explanation:

The expression $\frac{sin^5(x)}{x^5}$ can be rewritten as $\frac{1}{x^5}sin^5(x)$ .

Recall the Squeeze theorem can be used to solve for the limit. The sine function has a range from [0,1] , which means that the range must be inside this boundary.

$0\leq sin^5(x) \leq 1$

Multiply the $\frac{1}{x^5}$ term through.

$0\left(\frac{1}{x^5}\right)\leq \left(\frac{1}{x^5}\right)sin^5(x) \leq 1\left(\frac{1}{x^5}\right)$

Take the limit as approaches infinity for all terms.

$\lim_{x \to \infty}0\leq \lim_{x \to \infty}\frac{sin^5(x)}{x^5} \leq \lim_{x \to \infty}\frac{1}{x^5}$

$0\leq \lim_{x \to \infty}\frac{sin^5(x)}{x^5} \leq 0$

Since the left and right ends of this interval are zero, it can be concluded that $\lim_{x \to \infty}\frac{sin^5(x)}{x^5}$ must also approach to zero.

The correct answer is 0.

Report an Error

Example Question #1 : Limit Concepts

Determine the limit. $\lim_{x \to 0}-\frac{1}{x^2}$

Possible Answers:

$\infty$

$-\infty$

Correct answer:

$-\infty$

Explanation:

To determine, $\lim_{x \to 0}-\frac{1}{x^2}$ , graph the function $y=-\frac{1}{x^2}$ and notice the direction from the left and right of the curve as it approaches x=0 .

Both the left and right direction goes to negative infinity.

The answer is: $-\infty$

Report an Error

Example Question #2 : Limit Concepts

Which of the following is true?

Possible Answers:

If $\small \small \lim _{x\to a} f(x)+g(x)$ exists, then $\small \lim _{x\to a} f(x)$ and $\small \lim _{x\to a} g(x)$ both exist.

$\small \lim _{x\to a} f(x)$ and $\small \lim _{x\to a} g(x)$ exist if and only if $\small \small \lim _{x\to a} f(x)+g(x)$ exists.

If $\small \lim _{x\to a} f(x)$ and $\small \lim _{x\to a} g(x)$ , then $\small \lim _{x\to a} f(x)+g(x)$ exists.

If neither $\small \lim _{x\to a} f(x)$ nor $\small \small \lim _{x\to a} g(x)$ exist, then $\small \lim _{x\to a} f(x)+g(x)$ also doesn't exist.

Correct answer:

If $\small \lim _{x\to a} f(x)$ and $\small \lim _{x\to a} g(x)$ , then $\small \lim _{x\to a} f(x)+g(x)$ exists.

Explanation:

If $\small \lim _{x\to a} f(x)$ and $\small \lim _{x\to a} g(x)$ , then $\small \lim _{x\to a} f(x)+g(x)$ exists.

This can be proven rigorously using the $\small \epsilon-\delta$ definition of a limit, but it is most likely beyond the scope of your class.

Report an Error

Example Question #1 : Limits

Determine the limit: $\lim_{x \to \infty}\frac{3x^n}{9n!}$

Possible Answers:

$\frac{1}{3}$

$\infty$

Correct answer:

Explanation:

Isolate the constant in the limit.

$\lim_{x \to \infty}\frac{3x^n}{9n!} = \frac{1}{3}\lim_{x \to \infty}\frac{x^n}{n!}$

The limit property $\lim_{x \to \infty}\frac{x^n}{n!}=0$ .

Therefore:

$\frac{1}{3}\lim_{x \to \infty}\frac{x^n}{n!} = \frac{1}{3}(0)=0$

Report an Error

Example Question #6 : Limit Concepts

Evaluate the limit, if possible: $\lim_{x\to \infty}tan^{-1}(x^2)$

Possible Answers:

$\frac{\pi}{4}$

$\infty$

$\frac{\pi}{2}$

Correct answer:

$\frac{\pi}{2}$

Explanation:

To evaluate $\lim_{x\to \infty}tan^{-1}(x^2)$ , notice that the inside term x^2 will approach infinity after substitution. The inverse tangent of a very large number approaches to $\frac{\pi}{2} \textup{ radians}$ .

The answer is $\frac{\pi}{2}$ .

Report an Error

Example Question #1 : Limit Concepts

Evaluate the following limit:

$\lim_{x\rightarrow \infty }\frac{x^2+1}{3x^2}$

Possible Answers:

$\frac{1}{3}$

$-\infty$

$\infty$

$-\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

The first step is to factor out the highest degree term from the polynomial on top and bottom (essentially pulling out 1):

$\lim_{x\rightarrow \infty }\frac{x^2}{x^2}\left(\frac{1+\frac{1}{x^2}}{3}\right)$

which becomes

$\lim_{x\rightarrow \infty }\left(\frac{1+\frac{1}{x^2}}{3}\right)$

Evaluating the limit, we approach $\frac{1}{3}$ .

Report an Error

Example Question #2 : Limits

Evaluate the following limit:

$\lim_{x\rightarrow \infty }\frac{x}{2x^2+3x+1}$

Possible Answers:

$\frac{1}{3}$

$-\infty$

$\infty$

Correct answer:

Explanation:

To evaluate the limit, first pull out the largest power term from top and bottom (so we are removing 1, in essence):

$\lim_{x\rightarrow \infty }\frac{x^2(\frac{1}{x})}{x^2(2+\frac{3}{x}+\frac{1}{x^2})}$

which becomes

$\lim_{x\rightarrow \infty }\frac{(\frac{1}{x})}{(2+\frac{3}{x}+\frac{1}{x^2})}$

Plugging in infinity, we find that the numerator approaches zero, which makes the entire limit approach 0.

Report an Error

View Calculus 2 Tutors

Mayowa
Certified Tutor

Logan University, Bachelor of Science, Biology, General.

View Calculus 2 Tutors

Muhammad
Certified Tutor

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

View Calculus 2 Tutors

Dawn
Certified Tutor

University of Oregon, Bachelor in Arts, Biochemistry and Molecular Biology. Johns Hopkins University, Doctor of Philosophy, E...

All Calculus 2 Resources

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

Popular Subjects

Reading Tutors in Denver, Physics Tutors in Washington DC, GRE Tutors in Atlanta, GRE Tutors in San Diego, LSAT Tutors in Denver, Algebra Tutors in Houston, Reading Tutors in Houston, MCAT Tutors in Los Angeles, SAT Tutors in New York City, Calculus Tutors in Seattle

Popular Courses & Classes

ACT Courses & Classes in Chicago, GRE Courses & Classes in Philadelphia, SSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Denver, GRE Courses & Classes in Phoenix, Spanish Courses & Classes in San Diego, Spanish Courses & Classes in Boston, MCAT Courses & Classes in Los Angeles, ACT Courses & Classes in Boston, SSAT Courses & Classes in Washington DC

Popular Test Prep

ACT Test Prep in Chicago, GRE Test Prep in Denver, SSAT Test Prep in Atlanta, GMAT Test Prep in Atlanta, SAT Test Prep in Phoenix, GMAT Test Prep in Los Angeles, LSAT Test Prep in Philadelphia, SAT Test Prep in Denver, LSAT Test Prep in Atlanta, SSAT 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

Calculus 2 : Limits

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Limits

Example Question #1 : Limit Concepts

Example Question #2 : Limits

Example Question #2 : Limit Concepts

Example Question #1 : Limit Concepts

Example Question #2 : Limit Concepts

Example Question #1 : Limits

Example Question #6 : Limit Concepts

Example Question #1 : Limit Concepts

Example Question #2 : Limits

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: