Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Finding Limits and One-Sided 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 #441 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1+}\frac{-38\cdot (x - 1)^{3}}{13sin(3\cdot \pi x)^{3}}\end{align*}$

Possible Answers:

$-\infty$

$\infty$

$\frac{38}{(351\cdot \pi ^{3})}$

$-\frac{38}{(351\cdot \pi ^{3})}$

Correct answer:

$\frac{38}{(351\cdot \pi ^{3})}$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{-38\cdot (x - 1)^{3}}{13sin(3\cdot \pi x)^{3}}\rightarrow\frac{0}{0}\\&\frac{-114\cdot (x - 1)^{2}}{117\cdot \pi cos(3\cdot \pi x)sin(3\cdot \pi x)^{2}}\rightarrow\frac{0}{0}\\&\frac{228 - 228x}{702\cdot \pi ^{2}cos(3\cdot \pi x)^{2}sin(3\cdot \pi x) - 351\cdot \pi ^{2}sin(3\cdot \pi x)^{3}}\rightarrow\frac{0}{0}\\&\frac{-228}{2106\cdot \pi ^{3}cos(3\cdot \pi x)^{3} - 7371\cdot \pi ^{3}cos(3\cdot \pi x)sin(3\cdot \pi x)^{2}}\rightarrow\frac{-228}{-2106\cdot \pi ^{3}}\\&lim_{x\rightarrow1+}\frac{-38\cdot (x - 1)^{3}}{13sin(3\cdot \pi x)^{3}}=\frac{38}{(351\cdot \pi ^{3})}\end{align*}$

Report an Error

Example Question #441 : Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{-13\cdot (x - 1)^{2}}{-41cos(\frac{(7\cdot \pi x)}{2})}\end{align*}$

Possible Answers:

$-\frac{52}{(82369\cdot \pi ^{2})}$

$\frac{52}{(82369\cdot \pi ^{2})}$

$-\infty$

$\infty$

Correct answer:

Explanation:

Report an Error

Example Question #443 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0-}\frac{7tan(11x)^{2}}{29x^{3}}\end{align*}$

Possible Answers:

$-\frac{(847\cdot (-29)^{(\frac{1}{3})})}{29}$

$\frac{(847\cdot (-29)^{(\frac{1}{3})})}{29}$

$\infty$

$-\infty$

Correct answer:

$-\infty$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{7tan(11x)^{2}}{29x^{3}}\rightarrow\frac{0}{0}\\&\frac{14tan(11x)\cdot (11tan(11x)^{2} + 11)}{87x^{2}}\rightarrow\frac{0}{0}\\&\frac{308tan(11x)^{2}\cdot (11tan(11x)^{2} + 11) + 14\cdot (11tan(11x)^{2} + 11)^{2}}{174x}\rightarrow\frac{1694}{0}=-\infty\\&lim_{x\rightarrow0-}\frac{7tan(11x)^{2}}{29x^{3}}=-\infty\end{align*}$

Report an Error

Example Question #444 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0+}\frac{33sin(17\cdot \pi x)}{35ln(x + 1)^{3}}\end{align*}$

Possible Answers:

$\frac{(561\cdot 35^{(\frac{2}{3})}\cdot \pi )}{35}$

$-\infty$

$-\frac{(561\cdot 35^{(\frac{2}{3})}\cdot \pi )}{35}$

$\infty$

Correct answer:

$\infty$

Explanation:

Report an Error

Example Question #445 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0-}\frac{23x}{-5sin(18\cdot \pi x)}\end{align*}$

Possible Answers:

$-\frac{23}{(90\cdot \pi )}$

$\infty$

$\frac{23}{(90\cdot \pi )}$

$-\infty$

Correct answer:

$-\frac{23}{(90\cdot \pi )}$

Explanation:

Report an Error

Example Question #446 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0-}\frac{26sin(19\cdot \pi x)}{21cos(5\cdot \pi x -\frac{ \pi }{2})^{2}}\end{align*}$

Possible Answers:

$\infty$

$-\infty$

$-\frac{(494\cdot \pi )}{(525\cdot \pi ^{2})^{(\frac{1}{3})}}$

$\frac{(494\cdot \pi )}{(525\cdot \pi ^{2})^{(\frac{1}{3})}}$

Correct answer:

$-\infty$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{26sin(19\cdot \pi x)}{21cos(5\cdot \pi x -\frac{ \pi }{2})^{2}}\rightarrow\frac{0}{0}\\&\frac{494\cdot \pi cos(19\cdot \pi x)}{-210\cdot \pi cos(5\cdot \pi x -\frac{ \pi }{2})sin(5\cdot \pi x -\frac{ \pi }{2})}\rightarrow\frac{494\cdot \pi}{0}=-\infty\\&lim_{x\rightarrow0-}\frac{26sin(19\cdot \pi x)}{21cos(5\cdot \pi x -\frac{ \pi }{2})^{2}}=-\infty\end{align*}$

Report an Error

Example Question #447 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{46sin(19\cdot \pi x)^{2}}{19ln(x^{5})}\end{align*}$

Possible Answers:

$\infty$

$-\frac{(46\cdot \pi ^{2})}{25}$

$-\infty$

$\frac{(46\cdot \pi ^{2})}{25}$

Correct answer:

Explanation:

Report an Error

Example Question #451 : Calculus Ii

Evaluate the 1-sided limits

$\small \lim_{x->4^+}\ln(4x^2-16x)$

Possible Answers:

$\small -\infty$

$\small \infty$

Correct answer:

$\small -\infty$

Explanation:

$\small \lim_{x->4^+}\ln(4x^2-16x)$

If we were to look at a plot of the function, the limit would be obvious. Drawing the graph would be difficult without a graphing device. We can overcome this by first defining a new variable and assessing how it behaves as the original variable approaches $\small x=4$ from the right.

Let,

$\small u = 4x^2 -16x$

$\small \lim_{x->4^+}u(x)=\lim_{x-4^+}(4x^2-16x)=0$

Therefore, $\small \small u ->0^+$ as $\small x->4^+$

$\small \small \small \lim_{x->4^+}\ln(4x^2-16x)=\lim_{u->0^+}\ln(u)$

We notice that the function in terms of $\small u$ has a graph that is much easier to draw from recollection since we readily know how the natural logarithm behaves. In the plot of the natural log, the y-axis is a horizontal asympote, and the function becomes infinite in the negative direction as becomes arbitrarily close to 0 from the right. Hence $\lim_{u->0^+}\ln (u) = -\infty$ .

$\small \small \small \lim_{x->4^+}\ln(4x^2-16x)=\lim_{u->0^+}\ln(u)= -\infty$

$\small \small \small \lim_{x->4^+}\ln(4x^2-16x)= -\infty$

Calc problem 4 plot

The graph above is more familiar. This is not the graph of the original function $\ln(4x^2-16x)$ , which is in terms of The plot above represents the function defined in terms of the variable we defined, .

Report an Error

Example Question #401 : Finding Limits And One Sided Limits

One-Sided Limits

Find

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

Possible Answers:

$-\infty$

$\infty$

Correct answer:

$\infty$

Explanation:

As gets closer to 3 (but remains larger than 3), then x-3 gets closer to 0 (but remains a small positive number).

The numerator, , gets closer to 6.

So, $\frac{2x}{x-3}$ is an arbitrarily large positive number.

Thus, we conclude that the limit is $\infty$ .

Report an Error

Example Question #401 : Finding Limits And One Sided Limits

One-Sided Limits

Find

$\lim_{x \to 3^-}\frac{2x}{x-3}$

Possible Answers:

$\infty$

$-\infty$

Correct answer:

$-\infty$

Explanation:

As gets closer to 3 (but remains smaller than 3), then x-3 gets closer to 0 (but remains a small negative number).

The numerator, , gets closer to 6. So, $\frac{2x}{x-3}$ is an arbitrarily large negative number.

Thus, we conclude that the limit is $-\infty$ .

Report an Error

View Calculus 2 Tutors

John
Certified Tutor

The University of Texas at Arlington, Bachelor of Science, Mathematics. Stanford University, Master of Science, Aerospace Eng...

View Calculus 2 Tutors

Bertha
Certified Tutor

California Institute of Technology, Bachelor's (in progress), Chemical Engineering.

View Calculus 2 Tutors

Sundeep
Certified Tutor

Indian Institute of Technology, Engineering in Computer Science, Electrical Engineering. New York University, Masters in Busi...

All Calculus 2 Resources

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

Popular Subjects

Statistics Tutors in Dallas Fort Worth, Algebra Tutors in Chicago, French Tutors in Chicago, Math Tutors in Washington DC, ISEE Tutors in Washington DC, SAT Tutors in Philadelphia, English Tutors in Seattle, SAT Tutors in San Francisco-Bay Area, English Tutors in Philadelphia, Chemistry Tutors in Phoenix

Popular Courses & Classes

GRE Courses & Classes in Denver, GMAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Denver, SSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Denver, ACT Courses & Classes in Seattle, GRE Courses & Classes in Philadelphia, GRE Courses & Classes in Miami, ACT Courses & Classes in Chicago

Popular Test Prep

MCAT Test Prep in Chicago, MCAT Test Prep in Denver, GRE Test Prep in Denver, SSAT Test Prep in Seattle, GRE Test Prep in Boston, GRE Test Prep in Phoenix, MCAT Test Prep in New York City, ISEE Test Prep in Boston, SAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Atlanta

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 : Finding Limits and One-Sided Limits

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #441 : Calculus Ii

Example Question #441 : Limits

Example Question #443 : Calculus Ii

Example Question #444 : Calculus Ii

Example Question #445 : Calculus Ii

Example Question #446 : Calculus Ii

Example Question #447 : Calculus Ii

Example Question #451 : Calculus Ii

Example Question #401 : Finding Limits And One Sided Limits

Example Question #401 : 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: