Limits & Continuity Questions and Answers

5 A graphing calculator is recommended If a rock is thrown upward on the planet Mars with a velocity 8 m s its height in meters t seconds later is given by y 8t 1 86t2 Round your answers to two decimal places a Find the average velocity in m s over the given time intervals 1 1 2 1 1 1 5 1 1 1 iv 1 1 01 v 1 1 001 m s m s m s m s m s b Use your answers from part a to estimate the instantaneous velocity of the rock in m s when t 1

Prove the statement using the definition of a limit 7 2x lim x 1 3 2x en 7 2 3 E Select But Given 0 we need Select such that if 0 x 1 6 then 7 2x Thus lim 3 by the definition of a limit ut 7 2x 3 x x 1 x 1 x 1 Select 2x 7 x So if we choose Select then 0 x 1 8

1 04 Points Sketch the graph of a function that satisfies all of the given conditions lim f x 7 lim f x 5 lim f x 5 f 6 6 f 3 4 X 3 O 5 DETAILS SCALCET9 2 XP 2 006 0 2 Submissions Used Submit Answer y 8 6 X 5 6 5 y 2 5 X 5 61 41 2 5 MY NOTES x

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval e4 3x 0 1 The equation e4 3x is equivalent to the equation f x e 4 3x 0 f x is continuous on the interval 0 1 f 0 Intermediate Value Theorem Thus there is a root of the equation e 4 3x in the interval 0 1 and f 1 Since Select 0 Select there is a number cin 0 1 such that f c

4 1 04 Points Find each x value at which is discontinuous and for each x value determine whether fis continuous from the right or from the left or neither x 2 X DETAILS SCALCET9 2 5 045 0 2 Submissions Used X 10 x 2 ifx 0 Isf continuous from the right left or neither at this value O continuous from the right O continuous from the left Oneither if 0 x 1 if x 1 8 X smaller value f x ex larger value Is f continuous from the right left or neither at this value O continuous from the right O continuous from the left Oneither Sketch the graph of f 5 y 10 5 5 10 5 10 X 10 5 DO y 10 5 5 10 5 10 X 10 DO 5 y 10 5 5 10 5 10 X 10 DO 5 y 10 5 5 10F 5

6 7 8 4 S Domain Range Is it a function Domain Range Is it a function Domain Range Is it a function Pag

flect Back 2 points In each row of the following table you are given some information about behavior of a function f on an interval a b what if anything can you say about f on the ne interval Pick from the following choices can t say anything positive negative zero increasing creasing constant concave up concave down linear If f x is then f x is positive negative increasing decreasing constant concave up concave down linear

Problem 3 Find the domain of the following function f x In 9 x 8 A 8 B 8 C 9 73 D 9 73 E 8 89 F 0 89 G 0 89 H

Complete parts a through c below a A warehouse is 60 yards long 28 yards wide and 5 yards high What is the area of the warehouse floor If the warehouse is filled to half its height with tightly packed boxes what is the volum of the boxes b A room has a rectangular floor that measures 29 feet by 15 feet and a flat 8 foot ceiling What is the area of the floor and how much air does the room hold c A grain silo has a circular base with an area of 136 square feet and is 23 feet tall What is the total volume a The area of the warehouse floor is

Given the graph of f x above find the following a Domain b Range c Interval s on which f x is increasing d Interval s on which f x is decreasing e Interval s on which f x is constant Note You can earn partial credit on this problem 18 5 10 18 x

Given the graph of f x above find the following not use the U sign Use parentheses for incre a Domain b Range c Interval s on which f x is increasing d Interval s on which f x is decreasing e Interval s on which f x is constant 10 10 c 10 5

20 0 12 Points DETAILS SCALCET9 2 XP 5 015 Find each x value at which is discontinuous and for each x value determine whether fis continuous from the right or from the left or neither a smaller value X f x Oneither b larger value X O continuous from the right O continuous from the left x 4 lfxs1 1 x x 2 2 if 1 x 2 lfx22 O continuous from the right O continuous from the left Oneither 6 c Sketch the graph of f Select the correct graph f x 2 4 6 2 f x 6 2 4 6 X 2 f x 6 2 2 4 6 2 f x 6 2 4 6

5 0 11 Points The toll T charged for driving on a certain stretch of a toll road is 5 except during rush hours between 7 a m and 10 a m and between 4 p m and 7 p m when the toll is 7 a Sketch a graph of T as a function of the time t measured in hours past midnight T T O DETAILS t FFW 7 10 16 19 jump infinite Onone T is continuous 24 t DO Classify the discontinuities as removable jump or infinite O removable 7 7 10 16 19 24 T b Discuss the discontinuities of of this function Enter your answers as a comma separated list If an answer does not exist enter DNE 7 5 7 10 16 19 24 t O T Discuss the significance of the discontinuities of this function to someone who uses the road O Because of the sudden jumps in the toll drivers may want to avoid the higher rates between t 0 and t 7 between t 10 and t 16 and between t 19 and t 24 if feasible Because of the sudden jumps in the toll drivers may want to avoid the higher rates between t 7 and t 10 and between t 16 and t 19 if feasible O Because of the steady increases and decreases in the toll drivers may want to avoid the highest rates at t 7 and t 24 if feasible 7 10 90 16 19

x x 7 0 1 2 f x x x 7 is Select on the closed interval 1 2 f 1 in the specified interval and f 2 Since 5 11 there is a number c in 1 2 such that f c by the Intermediate Yalue Theorem

If f is continuous on co co what can you say about its graph Select all that apply The graph of f has a hole The graph of f has a jump The graph of f has a vertical asymptote Onone of these

3 0 11 Points Sketch the graph of a function f that is continuous except for the stated values s Discontinuous but continuous from the right at 5 Need Help DETAILS SCALCET9 2 XP 5 002 Submit An Read It Watch It y DO O MY NOTES ASK YOU

2 0 11 Points DETAILS SCALCET9 2 5 003 MI Use the graph to determine the x values at which is discontinuous For each x value determine whether fis continuous from the right from the left or neither Enter your answers from smallest to largest smallest value x O continuous from the right continuous from the left Oneither X 4711 2 2 O continuous from the right O continuous from the left Oneither O continuous from the right O continuous from the left O neither largest value X O continuous from the right continuous from the left 2 6 X

6 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a f x x 4x lim f x X111 lim X 11 Find f 1 F 1 lim lin lim x 1 lim x lim x 1 x 1 144 a 1 D D x lim x by the power law by the sum law by the multiple constant law by the direct substitution property Thus by the definition of continuity fis continuous at a 1

9 Let g 1 t 9 Vt 3 a Make two tables one showing values of g for t 8 9 8 99 and 8 999 and one showing values of g for t 9 1 9 01 and 9 001 t 9 9 Vt 3 b Make a conjecture about the value of lim

67 74 Identities Prove the following identities 67 sec 0 1 cos 0 69 tan 0 1 sec 0 71 sec 20 csc 0 73 cos x cos x 68 tan 0 sin 0 csc 0 72 sec x 7 74 siny s 70 sim COS COS sec

Location of vertical asymptotes Let f x the following limits and find the vertical asymptotes of f Verify your work with a graph ing utility a lim f x 21 b lim f x 2 1 c lim f x 1 1 Untitled Graph 1 4x 3 2 1 X 10 Evaluate desmos

The little rep lim h 0 O f x cos x a cos h 1 h 4 O f x cos x a 0 O f x cos x a O f x cos x a O f x cos x a Need Help Submit Answer velocity 17 TU 3 IT 6 Read It f t 18 DETAILS Watch It 12 0 11 Points A particle moves along a straight line with equation of motion s f t where s is measured in meters and t in seconds Find the velocity and speed in m s when t 4 35 t 1 SCALCET9 2 7 036 m s

Use the given graph of f to find a number such that if x 1 then If x 11 0 2 8 1 2 1 0 8 f 0 7 1 1 1 X

Use the graph of g to find the value of each expression If an answer does not exist enter DNE y a b d lim g x e X 0 c lim 09 x lim g x x 0 lim g x X 2 lim g x x 2 f lim g x 4 2 2 N 4 9 X

10 22 Points Prove the statement using the e definition of a limit x 5 im 3x 4 5 Given 0 we need Select such that if 0 x 51 6 then 4 3x 5 Illustrate with a diagram y 5 6 DETAILS 5 5 6 Thus lim x 5 5 by the definition of a limit 75 5 5 5 5 5 x 5 6 5 Select But 4 x 5 x 1 x 51 x 51 Select v So if we choose 6 5 5 8 5 6 X y 5 6 O 5 5 6 75 5 5 y 5 5 choose Select then 0 x 51 5 5 5 6 5 6 9

Let f x 1 x x a Estimate the value of the limit lim 1 x x to five decimal places Does this number look familiar x10 b Illustrate part a by graphing the function y 1 x x 15 10 5 Need Help 10 5 y 10 15 5 10 15 no 15 10 5 15 10 5 5 10 15 5 10 15 X O 15 10 5 16 10 5 10 15 5 10 15 X 15 10 5 y 15 10 5 5 10 15 MY 5

lim f x 6 lim f x 4 lim x 4 5 5 1 X 5 57 3 Need Help Submit Answer y lim 6 2 Read it Watch It DETAILS 5 X O 5 y 6 4 2 5 X 5 y 6 4 4 0 1 Points SCALCET9 2 2 025 Use a table of values to estimate the value of the limit correct to three decimal places If you have a graphing device use it to confirm your result graphically sin 58 2 5

Evaluate the function f x at the given numbers Round your answers to six decimal places f 9 5 f 9 1 f 9 05 f 9 01 f 9 005 f 9 001 f 8 9 E f 8 95 f 8 99 f 8 995 f 8 999 f x x 9x x 9 5 9 1 9 05 9 01 9 005 9 001 8 9 8 95 8 99 8 995 8 999 x x 72 Guess the value of the limit of f x as x approaches 9 Round your answer to six decimal places If an answer does not exist enter D x2 9x lim x 9 x x 72

a lim f x x 1 1 0 1 Points For the function f whose graph is given state the value of each quantity if it exists If an answer does not exist enter DNE b c lim f x x 3 lim f x x 3 d lim f x x 3 DETAILS e f 3 ya 4 2 0 SCALCET9 2 2 005 2 4 X

For the function of whose graph is shown state the following If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE a lim f x 7 b lim f x c lim 144 2 4 6 2 x 0 f x Figure d lim f x x 6 Description x

6 0 1 Points DETAILS A patient receives a 150 mg injection of a drug every 4 hours The graph shows the amount f t of the drug in the bloodstream after t hours f t 300 150 Find lim f t and t 12 t 12 lim f t lim f t 6 12 4 8 lim f t t 127 mg 19 2 2 0 10 MI mg 12 16

0 13 Points A tank contains 9 000 L of pure water Brine that contains 10 g of salt per liter of water is pumped into the tank at a rate of 25 L min a Find the concentration of salt after t minutes in g L c t DETAILS g L b What happens to the concentration in g L as t o g L

X y y x x4 and estimating the asymptotes Enter your answers as comma separated lists If an answer does not exis

2 0 13 Points Sketch a graph of a function f that satisfies all of the given conditions lim f x 00 lim f x 0 lim f x 0 O DETAILS SCALCET9 2 XP 6 002 Need Help y Read It x 1 Watch II lim f x 00 lim f x 00 x ot X 0 y X 1 X 1 X 1

Let P and Q be polynomials with positive leading coefficients Consider the limit below If the limit is infinite enter o or o as appropriate If the limit does not otherwise exist enter DNE P x lim a Find the limit if the degree of P is less than the degree of Q b Find the limit if the degree of P is greater than the degree of Q

Find the limit If an answer does not exist enter DNE x 9x 20 X 4 lim X 4

Consider the following functions and calculate their limits as x a numerically by computing the values of the functions near x a Make sure you compute the limit from both sides of a and include a table showing your calculations f x cos x 1 5 f x x a 0 X 2 x x 21 tan 2x 4 x x 2 a 2 Let f x 5 and estimate f 2 numerically by using definition of derivative for small enough values of h Consider the function f x e 2x In x 5 and find the equation for the line tangent to f x at x 0 Plot both f x and the tangent line in one graph to show that your answer is correct Position of an object as a function of time is given by x 3t 2t 5t 1 cos 2t Find velocity and acceleration of this object Plot position velocity and acceleration of this

The graph of a function h is given range 6 5 4 3 2 1 1 2 3 X 654 IT 321 a Find h 2 h 0 h 2 and h 3 h 2 h 0 h 2 h 3 4 e Find the net chan 45 5 1 2 b Find the domain and range of h Enter your answers using interval nota domain 3 d Find the values of x for which h x 4 O 2 4 and 3 O 2 4 O 3 2 4 O 3 2 and 4 O 3 2 4 5 6 c Find the values of x for which h x 4 Enter your answers as a comma X

The graphs of fand g are given Use them to evaluate each limit if it exists If an answer does not exist enter DNE y y a lim f x g x b lim f x g x c lim f x g x f x x 3 g x d lim e lim lim x x x 2 Fl 13 y f x X y g x e

A graphing calculator is recommended Use the Squeeze Theorem to show that lim x2 cos 22 x 0 x 0 Illustrate by graphing the functions f x x g x x cos 22mx and h x x on the same screen Let f x x 2 g x x cos 22mx and h x x2 Then 2 cos 22x s x cos 22mx Since lim f x lim h x by the Squeeze Theorem we have lim X

Let x 1 f x x a Find the following limits If an answer does not exist enter DNE lim f x lim f x b Does lim f x exist O Yes No c Sketch the graph of f y 1 0 x 1 if x 1 x 2 if x 2 1 0 5 www 2 2 0 5 1 0 1 5 2 0 2 3 4 O y 1 3 1 3 y 4 1 0 0 5 0 5 1 0 1 5 2 0 2 3 4 1 DO y 2 1 2

11 0 11 Points The signum or sign function denoted by sgn is defined by 1 if x 0 if x 0 if x 0 a Sketch the graph of this function y O sgn x DETAILS SCALCET9 2 3 049 iii lim sgn x 0 1 2 11 lim sgn x 310 4 2 2 4 2 b Find each of the following limits If an answer does not exist enter DNE 1 lim sgn x x 0 4 y 4 2 4 2 y 4 2 2 2 4 X 4 2 y 4 2

If 3x 5 f x x 3x 4 for x 0 find lim f x x 3

Find the limit If an answer does not exist enter DNE t 36 2t 13t 6 lim t 6 Need Help Read It Watch It

The figure below shows contours of g x y on the region R with 9 x 15 and 4 y 10 Using Az Ay 2 find an overestimate and an underestimate for SR 9 x y dA Overestimate 4 Underestimate 1 10 5 11 15

Find the maximum value of f x y 2 y for z y 20 on the unit circle

lim f x lim f x lim f x x a L 0 M I

0 y 2 lim 1 1 z 0 x lim 848 y 1

8 Use the graph of y f x to answer the following questions show away to lis work sfy 5 4 3 2 506 a Determine the domain stumboro b Determine the x intercept s vollot svo S 21 11231 mot avio2 8

1 x 21 models the percentage of households with an interfaith marriage I x years after 1989 The formula N x 9 models the percentage of households in which 1 The formula 1 x 2 a person of faith is married to someone with no religion N x years after 1989 Answer parts a d below a In which years will more than 34 of households have an interfaith marriage After the year b In which years will more than 19 of households have a person of faith married to someone with no religion After the year c Based on your answers to parts a and b in which years will more than 34 of households have an interfaith marriage and more than 19 have a faith no religion marriage After the year d Based on your answers to parts a and b in which years will more than 34 of households have an interfaith marriage or more than 19 have a faith no religion marriage After the year S 16 2025