Limits & Continuity Questions and Answers

1 point Consider the function x X f x X Find the vertical asymptote s If there is more than one vertical asymptote give a list of the x values separated by commas If there is no vertical asymptote type in None 10x 21 7 X If this function has a horizontal asymptote give its y value If there is no horizontal asymptote type in None Find the y intercept Find the x intercept s If there is more than one x intercept give a list of the x values separated by commas

Determine the convergence or divergence of the series using any appropriate test from this chapter Identify the test use 00 cos n 20 n 1 converges by the Direct Comparison Test O diverges by the Alternating Series Test diverges by the Direct Comparison Test

00 n 0 a n converges if lim 7 8 True O False 2 Click here to view the transcript 1 n 1 an

Find the limit Let m and n represent arbitrary real numbers x m 2 x nx lim X 00 m n 2 mx

Use the Ratio Test to determine the convergence or divergence of the series If the Ratio Test is inconclusive determine the convergence or divergence of the series using other methods If you need to use co or co enter INFINITY or I respectively lim n 1 converges O diverges n 45 1 4

Find the limit T 1 lim X 9 e 9 8e

QUESTION 13 got your water for your tea boiling to 212 F You prefer to drink your tea at a temperature of 100 F If you set it aside in a room where the temperature is 72 F how long until you can drink it Use Newton s Law Cooling to answer the question The k value is known to be 0 118 relative to time in minutes kt 6 T

Approximate the following logarithms given log 2 0 3562 log 3 0 5646 a log 60 b log 106 c log 1 d 10 b 100 11 log 5 0 8271

Find an angle A that is coterminal with an angle measuring 470 where 0 A 360 Do not include the degree symbol in your answer For example if your answer is 20 you would enter 20 Provide your answer below

Use the graph of F shown on the right to find the given limit When necessary state that the limit does not exist lim F x X 1 Select the correct choice below and fill in any answer boxes in your choice OA lim F x Type an integer or a simplified fraction X 1

The average price of a ticket to a baseball game can be approximated by p x 0 03x 0 42x 6 69 where x is the number of years after 1991 and p x is in dollars a Find p 5 b Find p 15 c Find p 15 p 5 d Find P 15 p 5 15 5 and interpret this result a p 5 Simplify your answer b p 15 Simplify your answer c p 15 p 5 Simplify your answer p 15 p 5 15 5 Simplify your answer Which of the following is the appropriate interpretation of this result d OA The result tells us that the price of a baseball ticket in 2016 is this amount more than in 2006 B From 1996 to 2006 the price of baseball tickets increased by this amount per year OC This amount was the average price difference of a baseball ticket from 1996 to 2006 OD From 1996 to 2006 the average price for a ticket increased on average by this amount per year

Use sigma notation to write the sum 5 4 2 4 1 4 3 5 4 14

Determine the convergence or divergence of the series If you need to use co or 00 enter INFINITY or INFINITY respectively n 1 4n 1 lim no 3n 1 converges diverges 1 4n 1 3n 1 4 3

The series True O False 00 n 0 n 2 1 converges

lim 0 converges O diverges convergence or divergence of the series If you need to use coor co enter INFINITY or INFINITY respectively

1 Determine function f x x Power series for the

converges conditionally converges absolutely diverges Need Help Submit Answer nwi Read It 6 25 6 25 Points Determine whether the series converges absolutely or conditionally or diverges COS n Watch It n DETAILS PREVIOUS ANSWERS LARCALCET7 9 5 057

Find the average value of f on 0 16 y 1 0 t 8 12 4X

College students were given three choices of pizza toppings and asked to choose one favorite The following table shows the results Find the probability that the favorite topping is meat given that the student is senior 0 194 O 0 283 0 469 O 0 085

Consider the following graph y 5 4 3 2 1 1 2 3 4 5 6 X a Find the interval s on which f is increasing Enter your answer using interval notation b Find the interval s on which fis decreasing Enter your answer using interval notation c Find the open interval s on which fis concave upward Enter your answer using interval notation d Find the interval s on which fis concave downward Enter your answer using interval notation e Find the coordinates of the point s of inflection

Problem 28 Use L Hopital s rule to find the limit 3x lim x0 ex 1

Problem 38 Determine the limit lim x 0 3x 1 e3x x

Section 11 3 L Hopital s Rule Problem 26 Use L Hopital s rule to find the limit x 1 x1 ln x3 lim

F x 3x 2 for x 3 for x 3 OA X Choose the correct graph below Find lim F x lim F x lim F x X 3 X 3 x 3 Ay OB Ay Q D O C O D

Use the graph to answer these questions a Find lim f x and lim f x X 4 X 4 b Find limf x X 4 c Find 1 4 d Is f x continuous at x 4 Why or why not a Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x X 4 OB The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice Simplify your answer OA lim f x Simplify your answer X 4 OB The limit does not exist b Select the correct choice below and if necessary fill in the answer box to complete your choice OA limf x Simplify your answer X 4 OB The limit does not exist c Select the correct choice below and if necessary fill in the answer box to complete your choice OA 1 4 Simplify your answer OB The function is undefined at x 4 d Is f x continuous at x 4 Why or why not OA Yes f x is continuous at x 4 because f 4 exists OB Yes f x is continuous at x 4 because limf x 1 4 X 4 OC No f x is not continuous at x 4 because limf x does not exist F Ay 10 04 0 y

Use the graph to estimate the average rate of change of the percentage of new employees from 1994 to 2000 from 2000 to 2006 and from 1994 to 2006 ICCEI What is the average rate of change of the percentage of new employees from 1994 to 2000 per year Type an integer or a decimal rounded to two decimal places as needed What is the average rate of change of the percentage of new employees from 2000 to 2006 per year Type an integer or a decimal rounded to two decimal places as needed What is the average rate of change of the percentage of new employees from 1994 to 2006 per year Type an integer or a decimal rounded to two decimal places as needed E Percent 30 25 20 151 10 5 0 5 10 Employment Year

Determine whether the function shown is continuous over the interval 6 6 Is the given function continuous over the interval 6 6 Yes 6 4 6 4 y f x 2 4

Determine whether the function is continuous over the interval 6 6 Is the function continuous over the interval 6 6 6 6 z

he total number of words M t that a person can memorize in t minutes is shown in the graph elow Complete parts a and b to the right AM 40 30 20 10 0 24 28 16 24 8 12 32 28 8 16 24 36 28 32 3640 ote Graph not drawn to scale Q a Find the average rate of change of M as t changes from 0 to 8 from 8 to 16 from 16 to 24 from 24 to 32 from 32 to 36 The average rate of change of M as t changes from 0 to 8 is Simplify your answer The average rate of change of M as t changes from 8 to 16 is Simplify your answer words min The average rate of change of M as t changes from 24 to 32 is Simplify your answer words min The average rate of change of M as t changes from 16 to 24 is words min Simplify your answer words min words min The average rate of change of M as t changes from 32 to 36 is Simplify your answer b Explain the behavior of the average rates of change after 24 min OA The graph becomes vertical and therefore has no slope OB The graph becomes vertical and therefore has an undefined slope OC The graph becomes horizontal and therefore has an undefined slope D The graph becomes horizontal and therefore has a slope of zero

a Find a simplified form of the difference quotient and b complete the following table f x 2x f x h f x h b Fill in the table a X X 3 3333 f h 3 2 itt 3 1 3 0 1 0 01 2 1 0 1 0 01 f x h f x h f x h f x h Type int

The initial substitution of x a yields the form x 3x 154 lim x 11 X 121 x 3x 154 x 121 B The limit does not exist Simplify the function algebraically or use a table or graph to determine the limit If necessary state that the limit does not ex Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim X 11 COLLE Type an integer or a simplified fraction

Determine the value for c so that lim f x exists x 3 f x 1 3x c for x 3 C x 6 for x 3 The value of cis

In a particular city taxicabs charge passengers 2 20 for entering a cab and then 0 30 for each one fifth of a mile or fraction thereof traveled If x represents the distance traveled in miles then C x is the cost of the taxi fare where C x 2 20 if x 0 C x 2 50 if 0 x 0 2 and so on The graph of C is shown to the right Use the graph to find each of the following limits if it exists Select the correct choice below and if necessary fill in the answer box to complete your choice A lim C x x 0 2 Simplify your answer Do not include the symbol in your answer OB The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim C x X 0 2 Simplify your answer Do not include the symbol in your answer B The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice A lim C x X 0 2 Simplify your answer Do not include the symbol in your answer The limit does not exist Cat Cost of taxi in dollars 3 7 3 4 3 1 2 8 2 5 2 24 0 NE 0 4 0 6 0 8 Distance traveled in miles 0 2

In a particular city taxicabs charge passengers 3 20 for entering a cab and then 0 40 for each one fifth of a mile or fraction thereof traveled If x represents the distance traveled in miles then C x is the cost of the taxi fare where C x 3 20 if x 0 C x 3 60 if 0 x 0 2 and so on The graph of C is shown to the right Use the graph to find each of the following limits if it exists Select the correct choice below and if necessary fill in the answer box to complete your choice O A lim C x X 0 26 Simplify your answer Do not include the symbol in your answer B The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim C x x 0 26 Simplify your answer Do not include the symbol in your answer OB The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim C x x 0 26 Simplify your answer Do not include the symbol in your answer B The limit does not exist C Cost of taxi in dollars 5 6 5 2 4 8 44 4 3 69 3 24 0 N 0 2 0 4 0 6 0 8 Distance traveled in miles 1

Find the indicated limit Note that l H pital s rule does not apply to every problem and some problems will require more than one application of l H pital s rule Use o or co when appropriate 2e x lim x In 1 7ex Select the correct choice below and if necessary fill in the answer box to complete your choice 2e x lim OA x In 1 7e 11 11 Type an exact answer in simplified form B The limit does not exist

Solve 9e7x 17 Round your answer to the nearest tenth Do not include x in your answer If necessary include a leading 0 with an decimals less than 1 For example 0 5 instead of 5

Solve 11e 21 Round your answer to the nearest tenth Do not include x in your answer If necessary include a leading 0 with am decimals less than 1 For example 0 5 instead of 5

Suppoes He demand function for x thousand of a certain item is p 130720 where x 1 and pis in dollars d find the marginal revenue b find the revenue from the next thousand a demand of 8000 x 8 The revenue from the next thousand items when intens at Crype an integer or decimal sounded to two decimal places as needed

21 use the properties of limits to find the indicated limit lim x 7x 3 X 8 x 3x 4 option A O B 3 Trokre of Gay Tub

30 use a calculator to estimate the limit mique 5X on word not lim X 0 option A 0 0 0 2015 02 ond B D Does not exist and is not or 00 mumxan Lou GNT A

the given limit 22 use the properties of limits to find nove lim kino 7x 7x 3 type an integer or a simplified fraction 23 8x5

Let f x 1 x 3 if x 2 3x 9 if x 2 Calculate the following limits Enter DNE if the limit does not exist lim f x 4 x 2 lim f x 4 X 2 lim f x 4 x 2 f 2 3

Solve 6 3 3 Enter an exact answer or round your answer to the nearest tenth Do not include x in your answe

Buse algebra and the properties of limits as needed to find the given limit If the limit does not exist Say so lim X 9 X 7X 18 X 9 Select the correct choice below and if nec the answer box fill in A lim x 7X 18 Type integer or a simplified X 9 X 9 B the limit does not exist an Praction Hanya de

1 Assume x and y are function of t Evaluate dyldt xy x 12 dx 3 x 2 y 5 tb ophon A g B3 C 3 di 9

The graph below is the function f x 5 4 3 2 1 5 2 2 3 Find lim f x x 1 Find f 1 Find lim f x x 1 Find lim f x x 1 F

4x lim x 5 x 5 lim x 5 4x x 5 4x lim x 5 x 5 11

The graph of g is given below Notice this is the graph of the derivative of g not the graph of g itself Answer the following questions about the graph of g based on the graph of g above A State the x values over which g is positive B State the x values over which g is negative C State the x values over which g is zero D State the x values over which g is undefined

A function f x is said to have a removable discontinuity at x a if 1 f is either not defined or not continuous at x a 2 f a could either be defined or redefined so that the new function IS continuous at x a 2x2 6x 80 x 5 Show that f x has a removable discontinuity at x 5 and determine what value for f 5 would make f x Let f x continuous at x Must define f 5 5

Find the center and the radius of the sphere y z 2x 3z 0 The center of the sphere is Simplify your answers The radius of the sphere is Tyne an exact answer using radicals as needed