Limits & Continuity Questions and Answers

Calculate these limits If the limit is o or oo write infinity or infinity If the limit does not exist write DNE Hint Remember the first thing you check when you are looking at a limit of a quotient is the limit value of the denominator 1 If the denominator does not go to 0 you should be able to right down the answer immediately 2 If the denominator goes to 0 but the numerator does not you will have to check the sign 1 of the quotient from both sides if the limit is not one sided 3 If both the numerator and the denominator go to 0 you have to do the algebraic trick of rationalizing So group your limits into these three forms and work with them one group at a time a lim t pl 2 b lim t pl 2 c lim t pi 2 d lim t pl 2 e lim f lim t pi 2 9 lim h lim sin t sin 2 t 17cos t 2cos 2 t cost sin t pl 27 t pl 2 1 lim t pi 2 sin t sin 2 t 17cos 2 t 2cos 2 t t pi 2 k lim t pl 2 1 lim t pi 2 2t 17cos t 2cos 2 t sin t sin 2t 17cos t 2sin t cos t sin 2 t 17cos 2 t 2cos 2 t 31 sin t sin 2 t 17cos t 2sin 2 t cos t sin 2 t 17cos 2 t 2sin t cost V sin 2 cos t 17cos 2 t 2sin 2 t sin t sin 2 t 17cos t 2cos t sin t sin 2 t 17cos 2 t 2cos t cost sin 2t 17cos 2 t 2cos t 700

A function is defined on the interval 2 1 2 by this multipart rule 7x x a a f x a 0 15 b 3 if x 0 31 tan x 31 cotx if 0 x 2 Here a and b are constants Find a and b so that the function f x is continuous at x 0 b if 2 x 0 X

4 5 Points DETAILS SCALCET9 2 XP 5 004 Consider the following if x 0 if x 20 a Find the left hand and right hand limits at the given value of a lim f x X10 f x lim f x x ot a 0 b Explain why the function is discontinuous at the given number a Since these limits are Select lim f x Select c Sketch the graph of the function and fis therefore discontinuous at 0

Match the statement with one of the graphs The slope of the line is negative and its x intercept is positive y 4 2 2 2 y 4 2 2 2 4 4 X X 4 4 2 2 4 y 2 y 4 2 2 4 X X

if x 1 if x 1 2 x2 If1 xs 2 X 5 If x 2 a Evaluate each of the following if it exists If an answer does not exist enter DNE g x 1 lim g x x 1 ii lim g x X 1 ii g 1 lv lim g x x 2 v lim g x 124 X 2 vi lim g x b Sketch the graph of g 10 5 g x 10 5 5 10 fu 10 X 10 00 5 g x 10 5 5 10 5 10 X 00 10 5 g x 10 5 5 10 10 X 08 10 5 g x 10 10 5

Evaluate the limit if it exists If an answer does not exist enter DNE Hint Note the name of the question is double rationalize which you can see at the top right corner So you have rationalize the conjugate trick both the numerator and the denomina 12 x 2 24 x 4 lim x 8

1 6 Points Given that DETAILS lim f x 4 lim g x 2 lim h x 0 find each limit if it exists If an answer does not exist enter DNE a lim f x 3g x X 2 b lim g x c lim f x X 2 d lim 5f x x 2 g x e lim 9 x x 2 h x SCALCET9 2 3 001 f lim 9 x h x X 2 f x

Matt places 1 200 in an investment account earning an annual rate of 6 5 compounded continuously Determine how much money Matt will have after 15 years

Prove that lim x sin 0 by using the Squeeze Theorem X 0

72 Passing through 3 7 and 1 2

66 1 4 and 1 0 For the following exercises write the equation of the line satisfying the given conditions in slope intercept form 67 Slope 6 passes through 1 3 68 Slope 3 passes through 3 2 69 Slope passes through 0 4 70 Slope x intercept 8 71 Passing through 2 1 and 2 1

For the following exercises for each of the piecewise defined functions a evaluate at the given values of the independent variable and b sketch the graph 4x 3 x 0 x 1 x 0 95 f x x 3 x 0 94 f x 96 h x 97 g x f 4 f 0 f 2 4x 3 x 0 x x 1 x 5 4 x 5 3 x 29 4 x 2 3 0 2 2 h 0 h h 5 g 0 g 4 g 2

Find the position and velocity of an object moving along a straight line with the given acceleration initial velocity and initial positic a t cos at v 0 9 s 0 6 The velocity is v t Type an exact answer

The total stopping distance T of a vehicle is shown below where T is in meters and x is the speed in kilometers per hour T 0 39x 0 006x Approximate the change and percent change in total stopping distance as speed changes from x 25 to x 28 kilometers per hour Round the percent change in total stopping distance to one decimal place dT x m X dT T m jaarna

For the following exercises for each pair of functions find a f g x and b g f x Simplify the results Find the domain of each of the results 42 f x 3x g x x 5 43 f x x 4 g x 4x 1 44 f x 2x 4 g x x 2 45 f x x 7 g x x 3 46 f x x g x x 9 47 f x 2x18 x 48 f x x 11 g x x x 4

a Dylan teint un meuble en bois Il a besoin de 12 ml de teinture pour recouvrir 100 cm On s int resse la relation entre la superficie teindre et la quantit de teinture n cessaire b Pour l anniversaire de Liam sa m re a achet un g teau de 525 g qu elle veut partager de fa on gale entre tous les invit s On s int resse la relation entre le nombre d invit s et la quantit de g teau par personne

Explain in a few words what the difference would be in the graph of y 2x 1 versus y 2x 1

Simplify Assume that all expressions under radicals represent nonnegative numbers 15a 6 10a 15a 6 79

ven that g x X 8 x 3 b g 8 find each of the following c g 3 d g 15 25 e g x h

Use the table to evaluate the given compositions a h g 1 b g f 4 d g h f 1 e f f f 1 h g f h 1 g f h g 0 j f f h 1 c h h 2 f h h h 4 i g g g 3 a h g 1 b g f 4 c h h 2 d g h f 1 Simplify your answer e f f f 1 Simplify your answer f h h h 4 g f h g 0 h g f h 1 Simplify your answer Simplify your answer Simplify your answer Simplify your answer Simplify your answer Simplify your answer 1 g g 9 3 Simplify your answer j f f h 1 Simplify your answer X f x g x h x 1 3 0 3 4 0 1 2 1 1 2 a 4 3 0 2 1 0

Simplify the expression of 0 30 1 01 3 x h 4 3x 4 h

3 Compute the following limits h 3 h 9 h 9 b lim b lim cos x sin x X R

4 x Let f x x ifx 2 if 2 x 2 a 6 x ifx22 Sketch the graph of f 2 2 10 2 00 Use the graph to determine the values of a for which lim f x does not exist Enter your answers as a comma separated list O 2 3 2 O

In a study of anxiety a group of participants shows a mean score of 35 with a standard deviation of 5 What is the percentage if the score is between 36 and 42 A 11 998 B 22 998 C 33 998 D 44 998

a Make a table of the factorial function for n 1 2 3 4 5 b Graph these data points and then connect them with a smooth curve c What is the least value of n for which n 106 a Complete the table below 1 n f n n OA 140 b Choose the correct graph below Af n 0 1 0 6 2 2 n OB Af n 140 3 6 0 4 24 6 n O C Af n 140 5 120 0 0 6 OD Af n 140 0 0 6 n

Hal function is defined for positive integers as n n n 1 n 2 3 2 1 a Make a table of the factorial function for n 1 2 3 4 5 b Graph these data points and then connect them with a smooth curve c What is the least value of n for which n 106 a Complete the table below 1 n f n n 2 3 4 5

Use the table to evaluate the given compositions b g f 2 a h g 2 d g h f 1 g f h g 0 J f f h 0 e f f f 0 h g f h 2 a h g 2 b g f 2 c h h 3 d g h f 1 Simplify your answer e f f f 0 Simplify your answer f h h h 1 Simplify your answer g f h g 0 Simplify your answer h g f h 2 Simplify your answer 1 g g g 2 Simplify your answer j f f h 0 Simplify your answer Simplify your answer Simplify your answer Simplify your answer c h h 3 f h h h 1 I g g g 2 II X f x g x h x 3 2 1 1 0 1 1 3102 3 20 0 3 1 0 1 2 3 2 2 2

9 If f x In x then lim x 3 Im A B e f x f 3 x 3 C In 3 is D nonexistent

A 1 lim 0 sin x B C 0 D nonexistent

Given that lim f x 4 and lim g x 5 find the limit lim lim 1 x sin x f x g x lim 1 lim f x 2 g x g x

Identify the transformations of the graph of f x 1 x Then graph g x on the same coordinate plane as the graph of f x by applying the transformations on the asymptotic x 0 and y 0 to the reference points 1 1 and 1 1 Also state the domain and range of g x using inequalities g x 2 1

Determine which of the following graphs is the graph of the given function f x x 5x 6 A C 28 O N 7 10 W B D AN T

Calculate the composite functions fo g and go f and determine their domains 1 g x x 6 x 1 f g x Domain O x 0 Ox 0 g f x Ox20 O all real numbers Ox 0 Domain 4 O all real numbers Ox 0 Ox 0 Ox20 Ox 0

Evaluate the following limit a lim 2x 3 X 1 X L b lim c lim 818 x 7x 6 X 6 V x 3x 2x 1

Let f x be the following piece wise f x x 1 if 0 x 1 1 x if x 1 Show lim f x does not exist X 1 the function function without grap

A exponential function is of the form f x a b c If the graph of f x has a horizontal asymptote at y 4 which of the following is true about the parameters of f x The parameter a must be 4 a 4 The parameter b must be 4 b 4 The parameter c must be 4 c 4 The product of the parameter a and b must be 4 a b 4

Problem 4 Which of the following sequences converge 1 1 n i an n 6 Problem 4 ii an iii an cos 2n 1 2 iv an sin 2n 1 2 D 1 1 n n 8 A all of them B i and iii only C ii and iv only D ii and iii only E 1 only F G none of them H i and iv only V

Problem 1 a 1 an 1 an 8 To show that an is monotonic using mathematical induction which of the following would be the second step A Assume az 1 8 ak 2 8 and show that a az 1 B Assume a 2 8 a 1 8 and show that at 1 az C Assume ak 1 a and show that ax 2 8 ax 1 8 x 18 D Assume ak ak 1 and show that a 8 E Assume az 1 8 a 8 and show that az 1 ak F Assume a 8 a 1 8 and show that a a 1 ax 1 8 ax 2 8 G Assume a ak 1 and show that H Assume ak 1 az and show that a 1 8 8 G

Problem 6 Use the Comparison Theorem to determine which of the following integrals converge S arctanx dx SVS 2 dx 26 0 2 2 sinx x512 dx A ii and iii only B i and iii only C i and ii only D i only E none of them F ii only G iii only H all of them

oblem 4 Which of the following sequences converge 1 1 n 72 6 1 1 2 8 ii an iii an cos 2n 1 2 iv an sin 2n 1 2 A all of them B i and iii only C ii and iv only D ii and iii only E 1 only F G none of them H 1 and iv only

Problem 2 Suppose that a sequence is defined by san 2 an Problem 2 a 1 an 1 Assuming that an is convergent find its limit

Problem 1 Suppose that a sequence is defined by a 1 an 1 an 8 To show that a is monotonic using mathematical induction which of the following would be the second step Problem 1 A Assume az 1 8 a 2 8 and show that a ax 1 B Assume ax 2 8 a 1 8 and show that C Assume at 1 a and show that ax 2 8 D Assume a az 1 and show that ax 8 E Assume ax 1 8 a 8 and show that az 1 a F Assume a 8 ax 1 a 1 8 and show that a az 1 8 G Assume ak ak 1 and show that H Assume ak 1 a and show that az 2 8 az 8 ax 1 ax a 1 8 ak 1 8 az 1 8 M

oblem 4 Which of the following sequences converge 1 1 n n 6 i an 1 1 n 8 ii an iii an cos 2n 1 2 iv an sin 2n 1 2 A all of them B i and iii only C ii and iv only D ii and iii only E i only G none of them H i and iv only

roblem 3 Find the limit of the following sequences a a ln 5n 5 In 6n b a n sin 8 n If the sequence does not converge then write divergent Enter your answer symbolically as in these examples Problem 3 a Enter your answer symbolically If the sequenc then write div If the sequenc

a Express each of the following numbers as a product of prime factors in index notation 1 25 ii 150 2x3x5 FEEDEC 2 marks b If the L C M of x and 25 is 150 find two possible values of x by using your answers in a 4 marks SPANNE DATORSVARE

A plane is going to fly 300 miles at a planned speed of 530 mph The flight will have an average headwind of w miles per hour the entire time meaning the plane is flying directly against the wind The time T in hours of the flight is a function of the speed of the headwind w in mph and can be modeled by T w 300 530 w a What does T 110 mean in this situation b At what value of w does the graph have a vertical asymptote Explain how you know and what this asymptote means in the situation T travel time 10 A 8 4 2 Here is the graph of y T w 100 400 200 300 w headwind speed 500 600

Problem 4 Let f x x log x 1 a Find the local and global extrema of f x b Find the local and global extrema of f x in the interval 1 2 c Calculate f f x dx and the second derivative of f x

x x 2 Problem 3 Let f x 4 7 3 a Is this function continuous at x 1 b Find lim 1 f x c Can we define f 1 such that f x is continuous at 1 If yes what is the value

Determine lim f x for the function f x shown in the figure below 20 4 1 10 18 1 0 The limit as 4 is help limits 18

Given lim 5 f x 7 and lim 5 g x 3 evaluate If the limit does not exist enter DNE Limit lim 2 5 f x g x 7 f x