Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Calculus II

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 #351 : Calculus Ii

Evaluate the following limit:

$\lim_{x\rightarrow 4^+}f(x), f(x)=\left\{\begin{matrix} x^2+1, x\leq 4\\ \ln(x-4), x> 4 \end{matrix}\right.$

Possible Answers:

$\infty$

$-\infty$

Correct answer:

$-\infty$

Explanation:

To evaluate the limit, we must determine which side of the limit we are approaching. The plus sign "exponent" means that we are approaching 4 from the right, using numbers larger than 4. Therefore, we must evaluate the limit using the second part of the piecewise function; when we evaluate the limit, we get the limit as the natural logarithm approaches zero, which gives us $-\infty$ .

Report an Error

Example Question #352 : Calculus Ii

Evaluate the following limit:

$\lim_{x\rightarrow 3} \frac{\ln(3-x)}{2}$

Possible Answers:

$-\infty$

$-\frac{1}{3}$

$\infty$

Correct answer:

$-\infty$

Explanation:

To evaluate the limit, we simply substitute the x value we are approaching - 3 - into the limit. As the natural logarithm gets closer to zero, we reach $-\infty$ .

Report an Error

Example Question #353 : Calculus Ii

Find $\lim_{x\rightarrow \infty} \frac{4x}{x^2+5x}$

Possible Answers:

$\infty$

$-\infty$

Correct answer:

Explanation:

$\lim_{x\rightarrow \infty} \frac{4x}{x^2+5x}$

By L'Hopital's rule:

$\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}$

$\lim_{x\rightarrow \infty} \frac{4x}{x^2+5x}=\lim_{x\rightarrow \infty} \frac{4}{2x+5}=0$

Report an Error

Example Question #354 : Calculus Ii

Find $\lim_{x\rightarrow 5}\frac{{(x-5)}^5}{sin(x-5)}$

Possible Answers:

$\infty$

Correct answer:

Explanation:

$\lim_{x\rightarrow 5}\frac{{(x-5})^5}{sin(x-5)}$

By L'Hopital's Rule:

$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}$ if $\frac{f(x)}{g(x)}=\frac{0}{0}$

Since this limit gives us $\frac{0}{0}$ we take the derivative of the numerator and denominator:

$\lim_{x\rightarrow 5}\frac{{(x-5})^5}{sin(x-5)}=\lim_{x\rightarrow 5}\frac{{5(x-5})^4}{cos(x-5)}=0$

Report an Error

Example Question #355 : Calculus Ii

Find $\lim_{x\rightarrow \frac{\pi}{2}^{-}}tan(x)$

Possible Answers:

$-\infty$

$\infty$

Correct answer:

$\infty$

Explanation:

We know that $tan(\frac{\pi}{2})$ is undefined. However, an easier way to determine this limit is to break up tangent into component parts:

$\lim_{x\rightarrow \frac{\pi}{2}^{-}}tan(x)=\lim_{x\rightarrow \frac{\pi}{2}^{-}}\frac{sin(x)}{cos(x)}$

Since it's approaching $\frac{\pi}{2}$ from the left, cos(x) approaches from positive values and sin(x) approaches from positive values. Since that's the case,

$\lim_{x\rightarrow \frac{\pi}{2}^{-}}tan(x)=\lim_{x\rightarrow \frac{\pi}{2}^{-}}\frac{sin(x)}{cos(x)}=\frac{1}{0}=\infty$

Report an Error

Example Question #354 : Limits

$\lim_{x\rightarrow -3} \frac{x^2-9}{x+3}$

Possible Answers:

$\infty$

$-\infty$

Correct answer:

Explanation:

To subsitute the value into the function, we must first factor the function to remove the hole.

x^2-9=(x-3)(x+3) , using difference of squares.

Therefore, $\frac{(x-3)(x+3)}{(x+3)}=x-3$

Now, we can just substitute! -3-3=-6.

Report an Error

Example Question #311 : Finding Limits And One Sided Limits

$\lim{x\rightarrow \infty} \; \frac{3x^3+2x^2+x}{(x+2)(x-2)}$

Possible Answers:

$-\infty$

$\infty$

DNE

Correct answer:

$\infty$

Explanation:

Whenever dealing with infinity limits, we only need to consider the highest power of the numinator and denominator. Therefore, we need to foil the denominator first.

$\frac{3x^3+2x^2+x}{x^2-4}$ .

If we only consider the highest powers (everything else will be insignificant when plugging in huge numbers), we only have $\frac{3x^3}{x^2}=3x$ . If you plug infinity into this, you will be get out infinity.

Report an Error

Example Question #354 : Calculus Ii

$\lim_{x \rightarrow 2}\frac{x^2-4}{x+2}$

Possible Answers:

DNE

Correct answer:

Explanation:

This question is the best case scenario for limits. Since the denominator does not equal zero when we plug in the value, we can just substitute the value into the function!

$\frac{2^2-4}{(2+2)}=\frac{0}{4}=0$ .

Report an Error

Example Question #355 : Calculus Ii

Evaluating the following right sided limit:

$\lim_{x\rightarrow 1^{+}}{\frac{x}{lnx}}$

Possible Answers:

$-\infty$

Undefined

$\infty$

Correct answer:

$\infty$

Explanation:

To evaulate this one sided limit, it is first worht noting that the function is not defined at x=1, due to the fact that ln(1)=0 causing division by zero. Furthermore, the two-sided limit is not defined since x is decreasing for the interval (0,1) and (1,e). This tells us that from the right side when x becomes arbitarly close to 1 but less than e, the function bcomes increasingly large. Also, the function is not defined for all x<0, since ln(x) is undefined for all x<0. This allows us to conclude that for values of x greater than 1 but less than e the function is tending twoards infinity.

Alternatively, for values of x on the interval (1,e), ln(x) is between (0,1), therefore 1/lnx or x/lnx will become increasingly large.

Report an Error

Example Question #355 : Calculus Ii

Find the limit

$\lim_{x \to \infty}e^{\frac{3}{x}}$

Possible Answers:

DNE

Correct answer:

Explanation:

Substituting $x=\infty$

we find that

$e^{\frac{3}{\infty}}\approx e^0=1$

As such

$\lim_{x \to \infty}e^{\frac{3}{x}}=1$

Report an Error

View Calculus 2 Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

View Calculus 2 Tutors

Jeremy
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Physics.

View Calculus 2 Tutors

Magdy
Certified Tutor

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

All Calculus 2 Resources

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

Popular Subjects

Spanish Tutors in Chicago, ACT Tutors in New York City, MCAT Tutors in Boston, Physics Tutors in Atlanta, GMAT Tutors in Denver, ISEE Tutors in Boston, Biology Tutors in San Francisco-Bay Area, Computer Science Tutors in Dallas Fort Worth, Biology Tutors in San Diego, French Tutors in Philadelphia

Popular Courses & Classes

ISEE Courses & Classes in Houston, SSAT Courses & Classes in Phoenix, ACT Courses & Classes in Houston, MCAT Courses & Classes in Seattle, SSAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Miami, MCAT Courses & Classes in Denver, GRE Courses & Classes in Philadelphia, GMAT Courses & Classes in Houston, ISEE Courses & Classes in Los Angeles

Popular Test Prep

GMAT Test Prep in Los Angeles, ISEE Test Prep in Seattle, GRE Test Prep in New York City, SAT Test Prep in Boston, SSAT Test Prep in Chicago, ISEE Test Prep in Philadelphia, ACT Test Prep in Seattle, ACT Test Prep in Phoenix, MCAT Test Prep in Phoenix, MCAT Test Prep in Miami

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 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #351 : Calculus Ii

Example Question #352 : Calculus Ii

Example Question #353 : Calculus Ii

Example Question #354 : Calculus Ii

Example Question #355 : Calculus Ii

Example Question #354 : Limits

Example Question #311 : Finding Limits And One Sided Limits

Example Question #354 : Calculus Ii

Example Question #355 : Calculus Ii

Example Question #355 : Calculus Ii

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: