Limits & Continuity Questions and Answers

9 Use algebraic and interval notation to describe all possible values of a so that 2 81

10 Which of the following statements is true about the set of real numbers described by the interval 15 1 A This set contains only one point x 15 y 1 B This set includes all real numbers between 15 and 1 not including 15 and 1 C This set is equivalent to the set given by 14 1 D 15 is not included in the set but a 1 is included

8 If you spend more than 50 at Old Navy shipping is free Let s be the total amount spent at Old Navy Use algebraic notation to describe all possible amounts you can spend to qualify for free shipping

Given that g is the inverse function of f and f 3 4 and f 3 5 then g 4 O A B O C 3 5 O E 3 5 D 1 3

15 42 Find the limit or show that it does not exist 4x 3 x 5x 1 15 lim 3t t 3 t t 4t 1 17 lim 1118 rr 118 2 3r 19 lim 21 lim 818 3 4 x 2 x 2 xx 3x 7 16 lim 18 lim 1110 3x3 8x 2 xx 4x 5x 2 20 lim 61 1 15 t 9 21 22 lim 8118 u 1 2u 1 u 2

3 For the function whosc graph is given state the following a lim f x b lim f x x18 x118 c lim f x x 1 e The equations of the asymptotes YA 1 1 d lim f x x 3 X

If the graph of y X C asymptote x 2 then b cannot be equal to A 0 ax b B 7 O C 5 OD 12 O E 10 has a horizontal asymptote y 5 and a vertical

1 Explain in your own words the meaning of each of the following a lim f x 5 818 b lim f x 3 8118

If y 2 f g x then y O A f g x g x B g x g x O c 2f g x g x D 2 g x g x 2 g x g x O E Cannot be determined

If y 2f g x then y A f g x g x B f g x g x c 2f g x g x D 2 g x g x 2 g x g OE Cannot be determined

The graph of which of the following equations has y 1 as an asymptote O A y sin x B y In x 3 2x OC y 2x 13x 7 OD y O E y ex D y 1

limx o e In x x3 O A OB O C 1 OD O E 1

The absolute minimum value of f x x 3x 12 on the closed interval 2 4 occurs at x OA 4 OB 2 O C 0 O D 1 OE 2

4 The function 7 x has the value J 1 dy the expression 2x 1 y 1 dx The slope of the curve at any point is A Write an equation for the line tangent to the curve y f x at x 1 B Use the tangent line from part A to estimate f 0 9 C Use separation of variables to find an explicit or implicit formula for y f x with no integrals remaining

Pegion R is bounded by the x axis y axis x 3 and y A Find the area of region R x 1

If limx 3 f x 7 which of the following must be true 1 f is continuous at x 3 II f is differentiable at x 3 III f 3 7 A None B II only C III only OD I and III only OE I II and III

Suppose an experimental population of amoeba increases according to the law of exponential growth There were 100 amoeba after the second day of the experiment and 300 amoeba after the fourth day Approximately how many amoeba were in the original sample A 10 B Not enough information to determine C 5 D 33 E 71

5x 1 An equation of the line tangent to the graph of y 3 1 at the point where x 1 is A 9x 7y 53 B 2x 3y 13 O C 16x 13y 44 OD 7x y 0 E x 2y 1

The graph of y g x is shown below At which point is g g OA A B B O C C D D OE E LA B C D Click here for long description x

A rock is dropped off the edge of a cliff and its distance s in feet from the top of the cliff after t seconds is s t 16t Assume the distance from the top of the cliff to the water below is 256 ft Complete parts a and b below Time Interval Average Velocity its 0 of 1 a When will the rock strike the water The rock will strike the water at 4 seconds Simplify your answer b Let t a be the time when the rock strikes the water Make a table of average velocities and approximate the velocity at which the rock strikes the water a 1 a a 0 5 a a 0 1 a a 0 01 a a 0 001 a ft sec Type an integer or decimal rounded to three decimal places as needed Save

limx 9x x 7x4 x 12x A 9 OB 7 O C C 72 12 OD 7 O E Does not exist

dx ves the slope at any point on the graph of the f 2 8 A Write the equation of the tangent line to f x at point 2 8 B Write an expression for f x in terms of x C What is the domain of f x D What is the minimum value of f x What is the minimum E Using the axes provided sketch a slope field for the given differential equation at the nine point indicated

is increasing at a rate directly proportional to 3500 C t Also C 0 1000 and C 5 2000 A Find C t as a function of t only B Calculate C 10 C Find lim C r and explain its meaning 1 w D On the axes provided draw a graph showing the number of cougars as a function of time 4000 3000 2000

5 6 The graph of a function f is given a At what numbers a does limx a f x not exist b At what numbers a is f not continuous c At what numbers does lim f x exist but is not continuous at a n 5 e 6

Consider the position function s t 16t 104t representing the position of an object moving vertically along a line Sketch a graph of s with the secant line passing through 0 75 s 0 75 and 2 s 2 Determine the slope of the secant line and explain its relationship to the moving object A As 144 69 0 75 Q OB AS 144 69 0 75 2 ON The slope of the secant line is msec 60 Type an integer or a decimal What is the relationship of the slope to the moving object OA The slope is the average velocity of the object over the interval 69 144 OB The slope is the instantaneous velocity when t 2 C The slope is the maximum velocity of the object OD The slope is the average velocity of the object over the interval 0 75 2 144 69 0 75 2 G

The graph of f x is shown below along with the line y x C a y x y f x d Click here for long description O D d f x 0 da OE f c a OA f x is a one to one function B a f f a C f x has a domain of d x C Assuming that f x is defined by restricting f x to the domain a x b which of the following statements is false b

25 x Find the points of discontinuity of f For each of them give the number Cat which the discontinuity occurs and the values of both one sided limits Enter INF for oo INF for DNE for does not exist CND for cannot determine If you find more than one point of discontinuity enter them in order from smallest to largest If there are answer boxes left unfilled fill each of them with an x Point 1 C lim f x x C lim f x x CH Point 2 C lim f x I C lim f x x C Point 3 C lim f x I C lim f x

lim f x Find the indicated one sided limits of f and determine the continuity of f at the indicated point NOTE Type DNE if a limit does not exist You should also sketch a graph of y f x including hollow and solid circles in the appropriate places I 6 lim f x I 6 f x lim f x I 6 6 Is f continuous at x 6 YES NO 4 x x 6 5 x x 6

THE RE HERE REFER MEREADERE WASSERANSE ABDGECACE CHARAC 2 If f is continuous on what can you say about its graph Er ARAPRAKINIM

Does this function have a discontinuity at x 0 Yes f x x X No

Solving the equation 16 x OA x 3 5 OB X 2 23607 OC x 1 or x 2 OD x 2 7182818 OE None of the above you get x

Which of the following ould represent an y f x A b where b 1 I II III 2 40T A I only B II only C III only D I and II only 30 20 10 2 x 2 1 20 1 y f x 40 X 2 2 5 5 1 1 4 y f x 2 2 5 5 20 2 40

For all positive x log log4 a Hint Think change of base O A log4 x log x B log4 2 1 2 O c log4 x O D log24 2 O E log x

Suppose y f x 3x 5 Then the inverse function f x 3 A 2 5 3 OB 5 5 O c 5 3 O D 3y 5 E 3x 5 3 5

Suppose 4 O O O O A In 12 OB 3 O c log4 12 OD log12 2 O E 12 en x

Measurements of a lake s width were taken at 15 foot intervals as show 0 f x 0 Estimate 120 If x dx f x dx with n 4 using each of the following methods 0 A Left hand approximation 15 30 45 60 75 90 105 120 15 18 20 19 23 24 22 12 B Right hand approximation C Trapezoidal approximation

The domain of the function f x arccos 3x2 2 is O A all x in the interval 1 3 U 1 OB all x in the interval 0 1 OC all x in the interval O D all real numbers OE all x in the interval 1 1

4 A certain computer algorithm used to solve very complicated differential equations uses an iterative method That is the algorithm solves the problem the first time very approximately and then uses that first solution to help it solve the problem a second time just a little bit better and then uses that second solution to help it solve the problem a third time just a little bit better and so on Unfortunately each iteration each new problem solved by using the previous solution takes a progressively longer amount of time In fact the amount of time it takes to process the k th iteration is given by T k 1 2k 1 seconds I A Use a definite integral to approximate the time in hours it will take the computer algorithm to run through 60 iterations Note that T k is the amount of time it takes to process just the k th iteration Explain your reasoning B The maximum error in the computer s solution after k iterations is given by Error 26

52 Let f x x 1 if x 1 x 2 if x 1 a Find limx f x and limx f x b Does limx 1 f x exist c Sketch the graph of f

11 34 Evaluate the limit if it exists 11 lim 3x 7 x 2 13 lim t 4 1 2t 8 t 4 15 lim x 2 17 lim x2 5x 4 x 2 x x 6 x2 3x 5x 2 12 lim 8 x x 6 14 lim x 3 x x 12 16 lim x 4 X 18 lim x 5 x 3x 2 2 x 3x x 12 2x 9x 5 x 25

How many definite integrals would be required to represent the area of the region enclosed by the curves y cos sin and y cos xsin 2 and y 0 03z assuming you could not use the absolute value function OA 4 OB 3 O C 5 O D 1 OE 2

Given that lim f x 4 x 2 find the limits that exist If the limit does not exist explain why a lim f x 5g x x 2 c lim f x x 2 lim g x 2 x 2 g x x 2 h x e lim lim h x C x 2 b lim g x x 2 3f x x 2 g x d lim f lim g x h x x 2 f x

lim lim x 2 3 lim In x 1 I 30 lim PEY F x 3 34 lim In sin x D limp

29 40 Determine the infinite limit 29 lim I 31 lim x 33 lim In x 1 33

Below is a graph of two curves x y 3y the S shaped 5 in red Which definite integral should you use to compute the area enclosed by these two curves A OA O B O c 2 1 36192 OD 1 55800 0 Click here for long description 1 55800 1 36192 y y 3y 5 dy 1 55800 1136192 y4 y 3y 5 dy 1 55800 y y 3y 5 dy y y 3y 5 dy 1 36192 OE 1 55800 y y 3y 5 dy

Use limits to compute the derivative 2 f 8 where f x x 7 f 8 Simplify your answer

7 1 point Sally is running a sunblock factory and there is a machine to fill the sunblock bottles The machines can fill 150 bottles an hour and each bottle takes 20 ounces of sunblock If Sally runs the machines for 10 hours at least how many gallons will Sally need to fill all of the bottles 204 gallons 234 gallons 235 gallons 213 gallons Previous Next

ot 1 After t hours of operation an assembly line has assembled A t 20t C x 2890 70x power lawn mowers 0 st 10 Suppose that the factory s cost of manufacturing x units is C x dollars where a Express the factory s cost as a composite function of the number of hours of operation of the assembly line b What is the cost of the first 2 hours of operation a The factory s cost is BILLE

X 7 x Let g x g h t and h x x 4x 7 Calculate the function g h t

The system x 0 Y Z 0 x 10 y 20 x y 2 5 x 2y 18 with the objective function p 70x 82y has a maximum of i 1028 ii 1020 iii 700 iv 732 V None of the options