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 #371 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&\lim_{x\rightarrow1-}\frac{5\cdot (x - 1)^{2}}{2cos(\frac{(19\cdot \pi x)}{2})^{2}}\end{align*}$

Possible Answers:

$-\frac{10}{(361\cdot \pi ^{2})}$

$-\infty$

$\frac{10}{(361\cdot \pi ^{2})}$

$\infty$

Correct answer:

$\frac{10}{(361\cdot \pi ^{2})}$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&\lim_{x\rightarrow n}\frac{f(x)}{g(x)}=\lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{5\cdot (x - 1)^{2}}{2cos(\frac{(19\cdot \pi x)}{2})^{2}}\rightarrow\frac{0}{0}\\&\frac{10x - 10}{-38\cdot \pi cos(\frac{(19\cdot \pi x)}{2})sin(\frac{(19\cdot \pi x)}{2})}\rightarrow\frac{0}{0}\\&\frac{10}{361\cdot \pi ^{2}sin(\frac{(19\cdot \pi x)}{2})^{2} - 361\cdot \pi ^{2}cos(\frac{(19\cdot \pi x)}{2})^{2}}\rightarrow\frac{10}{361\cdot \pi ^{2}}\\&\lim_{x\rightarrow1-}\frac{5\cdot (x - 1)^{2}}{2cos(\frac{(19\cdot \pi x)}{2})^{2}}=\frac{10}{(361\cdot \pi ^{2})}\end{align*}$

Report an Error

Example Question #371 : Finding Limits And One Sided Limits

Evaluate: $\lim_{x->3} \frac {x^4-81}{\sqrt {x^2-9}}$

Possible Answers:

Correct answer:

Explanation:

Screen shot 2016 01 01 at 9.27.10 pm

Report an Error

Example Question #373 : Finding Limits And One Sided Limits

$\lim_{x->\infty}\frac{x^4+2x-3}{3x^4+x^2-2 }$

Possible Answers:

$\frac{1}{3}$

$\frac{3}{2}$

Does not exist

$\frac{\infty}{\infty}$

Correct answer:

$\frac{1}{3}$

Explanation:

If you chose $\frac{\infty}{\infty}$ this is incorrect. Although both the numerator and denominator become infinite when you apply the condition $x->\infty$ , the $\frac{\infty}{\infty}$ is one of several indeterminate forms, meaning we cannot conclude anything about the behavior of the function as $x->\infty$ with the expression in this form.

Solution

$\lim_{x->\infty}\frac{x^4+2x-3}{3x^4+x^2-2 }$

$=\lim_{x->\infty}\frac{\frac{1}{x^4}}{\frac{1}{x^4}}\frac{(x^4+2x-3)}{(3x^4+x^2-2)}$

$=\lim_{x->\infty}\frac{(1+\frac{2}{x^3}-\frac{3}{x^4})}{(3+\frac{1}{x^2}-\frac{2}{x^4})}=\frac{1+0-0}{3+0-0}$

The terms in the numerator and the denominator will all vanish in the limit except for in the numerator and in the denominator. This means the limit is $\frac{1}{3}$ .

Alternate Solution 1

We could also have used L'Hoptitals rule which allows us to take derivatives of the numerator and denominator,

$\lim_{x->\infty}\frac{f(x)}{g(x)}=\lim_{x->\infty}\frac{f'(x)}{g'(x)}$ .

This operation can be performed successively until the limit is in determinant form.

$L=\lim_{x->\infty}\frac{(x^4+2x-3)'}{(3x^4+x^2-2)' }$

$=\lim_{x->\infty}\frac{(4x^3+2)'}{(12x^3+2x)' }$

$=\lim_{x->\infty}\frac{(12x^2)'}{(36x^2+2)' }$

$=\lim_{x->\infty}\frac{(24x)'}{(72x)' }$

$=\lim_{x->\infty}\frac{24}{72}$

$=\frac{1}{3}$

Alternate Solution 2 (The Quickest!)

Notice that the coefficients in front of the highest order polynomial terms in the numerator and denominator can be read off immediately to write the limit. This approach works when the highest order terms are raised to the same power in the numerator and denominator.

Report an Error

Example Question #374 : Finding Limits And One Sided Limits

Evaluate:

$\lim_{x\rightarrow 4+}\ln(x-4)-x^2$

Possible Answers:

$-\infty$

$\infty$

Correct answer:

$-\infty$

Explanation:

To evaluate the limit, we first must determine whether the limit is right or left sided; the positive sign exponent on 4 indicates that we are approaching 4 from the right side, using values slightly greater than 4. When we evaluate the limit, we see that the natural logarithm function goes to negative infinity as it approaches zero, and this dominates the behavior of the function overall (the polynomial is far smaller than infinity). So, our answer is $-\infty$ .

Report an Error

Example Question #372 : Finding Limits And One Sided Limits

Evaluate:

$\lim_{x\rightarrow \infty}\frac{11x^2+10x}{x+4}$

Possible Answers:

$\frac{5}{2}$

$-\infty$

$\infty$

Correct answer:

$\infty$

Explanation:

To evaluate the limit, we must factor out a term consisting of the highest power term divided by itself (so we aren't changing the function itself):

$\lim_{x\rightarrow \infty} (\frac{x^2}{x^2})\frac{11+10x^{-1}}{x^{-1}+4x^{-2}}$

All of the terms with negative exponents equal zero as the x approaches infinity (they are all fractions with infinitely large denominators), so we are left with our function with zero in its denominator, which equals infinity.

Report an Error

Example Question #373 : Finding Limits And One Sided Limits

Evaluate:

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

Possible Answers:

$\infty$

$-\infty$

Correct answer:

Explanation:

When evaluating the limit, we must look at the terms in the numerator and denominator. In the numerator, we have only polynomial terms. In the denominator, we have an exponential term, which grows faster than the terms in the numerator. Thus, the denominator grows faster than the numerator, so as x approaches infinity, the function goes to 0.

Report an Error

Example Question #374 : Finding Limits And One Sided Limits

Evaluate:

$\lim_{x\rightarrow \infty} \frac{2x}{5x+1}$

Possible Answers:

$\infty$

$\frac{2}{5}$

$-\infty$

Correct answer:

$\frac{2}{5}$

Explanation:

To evaluate the limit, we must factor out a term consisting of the highest power term divided by itself (so we aren't changing the function itself):

$\lim_{x\rightarrow \infty} (\frac{x}{x})(\frac{2}{5+x^{-1}})$

The factor we created cancels to become one, and the negative exponent term goes to zero (the fraction has an infinitely large denominator). What we are left with is $\frac{2}{5}$ .

Report an Error

Example Question #375 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{5tan(10\cdot \pi x)}{7ln(x^{6})}\end{align*}$

Possible Answers:

$\infty$

$\frac{(25\cdot \pi )}{21}$

$-\frac{(25\cdot \pi )}{21}$

$-\infty$

Correct answer:

$\frac{(25\cdot \pi )}{21}$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{5tan(10\cdot \pi x)}{7ln(x^{6})}\rightarrow\frac{0}{0}\\&\frac{50\cdot \pi \cdot (tan(10\cdot \pi x)^{2} + 1)}{\frac{42}{x}}\rightarrow\frac{50\cdot \pi}{42}\\&lim_{x\rightarrow1-}\frac{5tan(10\cdot \pi x)}{7ln(x^{6})}=\frac{(25\cdot \pi )}{21}\end{align*}$

Report an Error

Example Question #376 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1+}\frac{5ln(x^{2})^{2}}{sin(9\cdot \pi x)^{2}}\end{align*}$

Possible Answers:

$-\infty$

$\frac{20}{(81\cdot \pi ^{2})}$

$\infty$

$-\frac{20}{(81\cdot \pi ^{2})}$

Correct answer:

$\frac{20}{(81\cdot \pi ^{2})}$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{5ln(x^{2})^{2}}{sin(9\cdot \pi x)^{2}}\rightarrow\frac{0}{0}\\&\frac{\frac{(20ln(x^{2}))}{x}}{18\cdot \pi cos(9\cdot \pi x)sin(9\cdot \pi x)}\rightarrow\frac{0}{0}\\&\frac{\frac{40}{x^{2}}-\frac{ (20ln(x^{2}))}{x^{2}}}{162\cdot \pi ^{2}cos(9\cdot \pi x)^{2} - 162\cdot \pi ^{2}sin(9\cdot \pi x)^{2}}\rightarrow\frac{40}{162\cdot \pi ^{2}}\\&lim_{x\rightarrow1+}\frac{5ln(x^{2})^{2}}{sin(9\cdot \pi x)^{2}}=\frac{20}{(81\cdot \pi ^{2})}\end{align*}$

Report an Error

Example Question #377 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1+}\frac{-7ln(x^{8})^{2}}{23\cdot (x - 1)^{2}}\end{align*}$

Possible Answers:

$-\frac{448}{23}$

$\infty$

$-\infty$

$\frac{448}{23}$

Correct answer:

$-\frac{448}{23}$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{-7ln(x^{8})^{2}}{23\cdot (x - 1)^{2}}\rightarrow\frac{0}{0}\\&\frac{-\frac{(112ln(x^{8}))}{x}}{46x - 46}\rightarrow\frac{0}{0}\\&\frac{\frac{(112ln(x^{8}))}{x^{2}}-\frac{ 896}{x^{2}}}{46}\rightarrow\frac{-896}{46}\\&lim_{x\rightarrow1+}\frac{-7ln(x^{8})^{2}}{23\cdot (x - 1)^{2}}=-\frac{448}{23}\end{align*}$

Report an Error

View Calculus 2 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 2 Tutors

Samantha
Certified Tutor

Middlesex County College, Associate in Science, Sustainability Studies.

View Calculus 2 Tutors

Millat
Certified Tutor

Rensselaer Polytechnic Institute, Bachelor in Arts, Biomedical Engineering.

All Calculus 2 Resources

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

Popular Subjects

Biology Tutors in Phoenix, GRE Tutors in Phoenix, ISEE Tutors in Miami, French Tutors in Philadelphia, MCAT Tutors in San Diego, Biology Tutors in San Diego, Computer Science Tutors in Chicago, Chemistry Tutors in Boston, Chemistry Tutors in Miami, Chemistry Tutors in Houston

Popular Courses & Classes

SAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Phoenix, Spanish Courses & Classes in Washington DC, MCAT Courses & Classes in Atlanta, Spanish Courses & Classes in Denver, SSAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Seattle, ISEE Courses & Classes in Boston, SAT Courses & Classes in Boston

Popular Test Prep

LSAT Test Prep in Los Angeles, GRE Test Prep in Seattle, MCAT Test Prep in New York City, MCAT Test Prep in Dallas Fort Worth, GRE Test Prep in Dallas Fort Worth, LSAT Test Prep in Boston, ACT Test Prep in Phoenix, MCAT Test Prep in Washington DC, LSAT Test Prep in San Francisco-Bay Area, SAT Test Prep in Los Angeles

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 #371 : Finding Limits And One Sided Limits

Example Question #371 : Finding Limits And One Sided Limits

Example Question #373 : Finding Limits And One Sided Limits

Example Question #374 : Finding Limits And One Sided Limits

Example Question #372 : Finding Limits And One Sided Limits

Example Question #373 : Finding Limits And One Sided Limits

Example Question #374 : Finding Limits And One Sided Limits

Example Question #375 : Finding Limits And One Sided Limits

Example Question #376 : Finding Limits And One Sided Limits

Example Question #377 : 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: