Limits & Continuity Questions and Answers

Use the given graph of the function g to find the following limits 1 lim g z help limits 2 lim g z 3 lim g x 4 lim g z 2 10 103

If an arrow is shot upward on the moon with a velocity of 58 m s its height in meters after t seconds is given by y 58t 83t a Find the average velocity over the given time intervals 1 1 1 5 2 1 1 1 3 1 1 01 4 1 1 001 b Find the instantaneous velocity after one second to the nearest hundredth 1 2 3 4 b m s m s m s m s m s

Find in terms of the constant a Limit lim h

Evaluate the limit If the limit does not exist enter DNE Limit lim 24 7 x 16x 63 x 7

Find following limits if they exist Enter DNE if the limit does not exist 1 lim f x h x Pa 2 lim f x h x HIG 3 lim f x g x 216 4 lim rah x 5 lim ra g x g x 6 lim a f x 7 lim h x 2 1 8 lim zah x 1 zoh x g x lim f x 2 lim h x 6 lim g x 0 2 0 7 0 2 0 9 lim

Evaluate the limit Enter I for I for oo and DNE if the limit does not exist Limit lim x 4 7x 2

Evaluate the limit If the limit does not exist enter DNE Limit 0 49 lim 2 7 x 7

Fill in the table and guess the value of the limit lim f y where f y y 5y 6 y 3y 18 y 6 Y 6 002 6 001 6 0005 6 0001 f y Y 5 998 5 999 5 9995 5 9999 The limit as y 6 is f y

Use the given graph of the function f to find the following limits If a limit does not exist type DNE 1 lim f 2 2 12 2 lim f 2 3 lim f x 2 3 4 lim f a 20

Evaluate the limit If the limit does not exist enter DNE Limit lim 2 2 42 82 4

Given limx 3 g x 5 evaluate If the limit does not exist enter DNE Limit lim g x x 3

Let F be the function below If you are having a hard time seeing the picture clearly click on the picture It will expand to a larger picture on its own page so that you can inspect it more clearly Evaluate each of the following expressions Note Enter DNE if the limit does not exist or is not defined a lim F x lim F z lim F x F 1 2 1 11 b lim F x z 1 c lim F x z 3 lim F x lim F x 2 1 lim F z lim F x 2 31 F 1 F 3

lim n a W i 0 1 Is n 3 4 n

im 8 CO h i 0 S 3 tc 4 n

The function f is given by f x 5 0 7 Which of the following describes f B The function f models exponential decay and lim f x 0 The function f models exponential decay and lim f x 8 The function f models exponential growth and lim f x 0 z 00 The function f models exponential growth and lim f x Z 80

The functions f and g are given by f x and g x What is the domain of the function h given by h x f g x A B All real numbers All real numbers not equal to 0 All real numbers greater than 0 All real numbers greater than or equal to 0

The function f is given by x x 1 and the function g is given by g x 2 3 Which of the following is an expression for f g x I B D 2 37 2 3 7 2 6x49 1 1

4 The function k is given by k x 92 Which of the following expressions also defines k x B O 2 31 3 2x 3 32 3 2

The function fis defined by f x 4 for 2 x 0 Which of the following expressions defines f x B 4 for 2 x 0 4 2 for 2 x 0 4 for 0 x 2 4 2 for 0 x 2

Two functions f and g are composed such that f g x 4 x 3 2 Which of the following could be the expressions for f x and g x A B f x x and g x 4x 3 f x 4x 3 and g x f x x 3 and g x 4x 2 f x 4x 2 and g x x 3

The function fis given by f x 9 25 Which of the following is an equivalent form for f x B f x 3 5 2 f x 3 5 22 f x 9 5 2 2 f x 9 5 21

1 Find limits a 1 point b 1 point c 1 point d 1 point x 4 lim 22 2x 2x 12 lim 2 1 3 Find derivatives a 1 point using definition y x 5 b 1 point cos 2ry 5x 3y c 1 point y x inz 4 1 5 point You have a function x2 4x 20 5 e 2 1 lim 2 0 3r lim e x PIX 2 1 5 point What type of breakpoint function h r has in point 3 h x 23 if x 3 if r 3 f x 2 1 x 3 Find critical points of the function check which of them are minimum and which of them are maxim Find the intervals in which function is increasing and in which it is decreasing

Problem 2 1 point Evaluate the following limit 2x A 3 3 B 3 12 C D 4 lim 0 cos 3x 1 E 1 3

Find the indicated limit lim x In x3400 x 0 lim x Inx int 3400

A dairy is planning to introduce and promote a new line of organic ice cream After test marketing the new line in a large city the marketing research department found that the demand in that city is given approximately by the following equation where x thousand quarts were sold per week at a price of Sp each and whose revenue function is given as R x xp p 15 0 x 6 A Find the local extrema for the revenue function 8 On which intervals is the graph of the revenue function concave upward Concave downward A What is are the local maximum a Select the correct choice below and if necessary fill in the answer box to complete your choice OA The local maximum a is are at x Simplify your answer Use a comma to separate answers as needed OB There is no local maximum What is are the local minimumva Select the correct choice below and if necessary fill in the answer box to completeur choice OA The local minimum a is are at x Simplify your answer Use a comma to separate answer OB There is no local minimum swers as needed B On which intervals is the graph of the revenue function concave upward Select the correct choice below and if necessary fill in the answer box to complete your choice OA The graph is concave upward on

The function f x is continuous on 00 00 Use the given information to sketch the graph of f 1 0 20 1 10 0 1 20 20 f x 0 on 0 0 and 20 00 10 0 f x 0 on 10 00 Choose the correct graph off below O A O B f 0 0 f 20 0 f x 0 on 0 20 t x 0 on 0 10 2 O C O D

Sketch the parabola and line on the same graph y 3x 5 and y 2x 3 O V No Solution Help 3 2 25 24 23 22 21 20 19 18 17 16 15 14 13 12 14 10 9 8 7 6 S5 NE 4 3 2 1 42445 2 Find the area between them from x 0 to x 3 square units 2 Fill WebAssign Graphing Tool Graph La After you can use propertie

The graph of y f x is shown to the right Identify the x coordinates of the points where f x has a local minimum N a b e fg h o Which of the following shows every x coordinate where f x has a local minimum Choose the correct answer below OA b e g OB c f D c f h OC a c f

Consider the function f x derivatives are given by f x 16x x 4 x 4 x 4 The first and second 16 4 3x x 4 14 Find the horizontal asymptotes A y 2 B y 1 C y 0 D DNE 15 Find the intervals where the function is increasing A 2 3 2 3 B 0 2 3 C 0 D 0 0 16 Find the inflection points Lo LOG ENT 910 A 2 3 0 5 B 0 1 C 2 3 0 5 2 3 0 5 E DNE

cos x lima x n r

limx In x 1 In x

3 lim x x x 5 x x HINT Use the conjugate of x x 5 to simplify the expression DAUPHINS A Does not exist B 1 C 0 D 1 2 E 5

4 lim sin x sin x X T

Question 2 The half life of Radium 226 is 1590 years If a sample contains 300 mg how many mg will remain after 4000 years mg Give your answer accurate to at least 2 decimal places

k x e sin x

In x limx 0 sin a limx 2x 1 3x 1 32 22 limx 0x2

1 Calculate the following limits Use L Hopital s rule if necessary X e x cos x x limx In 1 limx 0 limx 1 sin x x e2 e e e 1 x 1 limx 0 lim 0 cos sin x 1 limx x 4x 3 2x 1

4 In a process bacteria are utilized to produce a specific enzyme through fermentation The rate of enzyme production E t measured in units per hour is modeled by the function E t 100t2t where t is the time in hours since the beginning of the fermentation process Determine the time at which the enzyme production is maximized during the process and find the maximum enzyme production rate

22 tan 945 X

10 Represent f x 1 2x as a power series What is its radius of convergence

f x x ax 15 x2 9 lim f x X 3 Find the limit

Refer to the graph to the right and follow the directions in parts a through f a Is a 1 or is 0 a 1 b Give the domain and range of f and the equation of the asymptote c Sketch the graph of g x ax d Give the domain and range of g and the equation of the asymptote e Sketch the graph of h x ax f Give the domain and range of h and the equation of the asymptote a Is a 1 or is 0 a 1 Choose the correct answer below 1 a f x ax 1 a 0 1 b Give the domain and range of f and the equation of the asymptoto OA For the function f x ax 0 a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis X OC For the function f x a a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis OB For the function f x a a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis OD For the function f x a 0 a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis Q G Ly

18 tan 0 36 cos 330

16 tan 120

40 Find m B A 29 km 128 12 km C B

Let g x y t at where is the function whose graph is shown 16 f 24 t q a Evaluate g x for x 0 4 8 12 16 20 and 24 g 0 g 4 g 8 g 12 9 16 g 20 g 24 b Estimate g 28 Use the midpoint to get the most precise estimate g 28

Oil leaked from a tank at a rate of r t liters per hour The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table Find lower and upper estimates for the total amount of oil that leaked out V V L lower estimate Question Help L upper estimate t h 0 2 4 6 8 10 r t L h 8 8 7 9 6 8 6 2 5 7 5 1

Is the function a polynomial function f x O O 4 X Yes No 9x 9

Use the graph of the function f to find lim f x and X 8 lim f x X 00 Select the correct choice below and fill in any answer boxes in your choice OA lim f x X OB The limit does not exist and is not co or 0 Select the correct choice below and fill in any answer boxes in your choice OA lim f x X 00 OB The limit does not exist and is not oo or 00 10 BB

8 Find the radius of convergence and interval of convergence of n 1 xn n5n