Limits & Continuity Questions and Answers

10 g t 1 In t

Evaluate the limit using L Hopital s rule lim 3 cos 52 sec 7x

a If the symbol x denotes the greates integer function defined in this example evaluate the following If an answer does not exist enter DNE i lim 12 ii lim x 2112 iii lim ain x b If n is an integer evaluate the following i lim ii lim 2 3 2 n c For what values of a does lim exist x The limit exists only for a 0 The limit exists for all non integer values of a The limit exists for all positive values of a

Compute the following limits using l Hospital s rule if appropriate Enter Inf if your answer is enter Inf if your answer is and en 32 3 z122 1 tan 2 lim zxx 1 2 3 11

Use continuity to evaluate the limit 40 9 40 x Consider the intervals for which the numerator and the denominator are continuous The numerator 40 is contintuous on the interval lim The denominator 40 is continuous and nonzero on the interval The interval for which the quotient is continuous is the intersection of the above intervals 40 Therefore the quotient 40 x Since 9 is in the interval 0 then f is continuous at x 9 Therefore 40 lim 1 9 40 x is continuous on the interval f 9

Evaluate the following limit Enter Inf if your answer is 2x ta lim z 0 5x ta enter Inf if your answer is oo and enter D

Show that the following limit is true 1 lim 240 First note that we cannot use 1 lim sin lim 2 0 I 2 0 sin I sin 1 lim sin I 0 1 because the limit as a approaches 0 of sin does not exist see this example Instead we a I 1 the Squeeze Theorem and so we need to find a function f smaller than g x x sin function h bigger than g such that both f x and h x approach 0 To do this we use our knowledge of the sine function Because the sine of any number lies between and we can write x 1 x and Any inequality remains true when multiplied by a positive number We know that a 0 for all a and so multiplying each side of inequalities of x we get 1

Evaluate the limit using L Hospital s rule if necessary 7100

Evaluate the limit using L Hospital s rule if necessary et lim 70 sin 4x

a What is wrong with the following equation 2 x 42 I 6 O x 6 x 7 x x 42 O The left hand side is not defined for x O The left hand side is not defined for x O None if these the equation is correct x 7 b In view of part a explain why the equation x x 42 x 6 lim 2 6 is correct Since x x 42 x 6 lim x 7 2 6 0 but the right hand side is 6 but the right hand side is and 7 are both continuous the equation follows O Since the equation holds for all x6 it follows that both sides of the equation approach th same limit as a 6 This equation follows from the fact that the equation in part a is correct None of these the equation is not correct

Evaluate the limit below 2 h 4 h lim h 0 If we define F h then we can t compute the limit by letting h 0 since F 0 is undefined But if we simplify E algebraically we find that F h lim h 0 2 h 4 h Recall that we consider only h 0 when letting h approach 0 Thus 2 h 4 h lim h 0 Question Help Message instructor A

lim f x 9 x 1 b lim g x x 1 find the limits if they exist If an answer does not exist enter DNE a lim f x 2g x x 1 c lim 2 1 4f x x 1 g x d lim f x g x 2 1 h x e lim f lim lim g x 4 2 1 g x h x f x lim h x 0 2 1

Evaluate the limit using the appropriate Limit Law s If an answer does not exist enter DNE t4 5 lim t 2 2t 3t 9 11 19 X Question Help Message instructor

Find the limit if it exists 6x 30 lim 2 5 x 5 Recall that x 5 x 5 if x 5 x 5 if x 5 Therefore we will need to check the limits when approaching from the left and from the right We will start by checking the limit when approaching from the left As a approaches 5 from left we have lim 2 5 6x 30 x 5 Next we check the limit when approaching from the right As a approaches 5 from the right w have 6x 30 lim 5 x 5 Since the left and right limits are Select an answer exist enter DNE 6x 30 lim 2 5 x 5 Question Help Video M the limit is as follows If an answer does not

Find the limit If an answer does not exist enter DNE sin 8 lim 0 0 70 tan 0

Suppose that Find f 1 f 1 Spy

Find the solution of the exponential equation in terms of logarithms or correct to four decimal places e 4 16

y 10 5 TE O O 40 O 5 0 5 Find the rule of the red portion of the function 10 5 10 powered by desmos Find the restricted domain of the red portion of the function Use interval notation

5 Determine the intervals of increasing myopenmath com Determine the intervals of decreasing If an interval does not exist write DNE Separate intervals using a union U Determine the intervals for which the function is continuous Determine the interval for which the function is differentiable Determine the behavior at x 4 continuous and not differentiable corner Determine the behavior at x 3 discontinous gap and not differentiable Determine the behavior at x 1 contiuous and not differentiable cusp Determine the intervals for which the function is concave upward Determine the intervals for which the function is concave downward 4 Determine if there are any points of inflection 2 0

Problem 2 Evaluate the following limit lim h 0 71 5 x h V71 5x h

Find the limit If needed enter Inf for co Inf for o or dne if the limit does not esist lim 1 023 1 100

Find the limit If needed enter Inf for Inf for or dne if the limit does not esist co 8 4 8 lim 00 2 9 8

Find the limit If needed enter Inf for Inffor o or dne if the limit does not esist lim e 00 cos 4x

Evaluate the following limits If needed enter Inf for Inf for or dne if the limit does not esist 5 A lim 2100 4 5 0 0

5 The graph of g is shown to the right Which of the following statements are true of g at x d 1 g is continuous 11 g is differentiable III g is increasing A only 1 B only III I and II only D II and III only E 1 II and III A 3 6 If f x is continuous on 1 3 and 3 f x 1 for all x in 1 3 then the greatest possible value of f 3 f 1 is 7 Let f x 8 If lim X4 2 B 1 flxl 2 x A 8 g is not increasing on this graph but it is differentiablo A Cormors sines the limit is the same coming from the Toft right 3x 10x 4x k x 4x 4 C 2 if x 1 if x 1 A f is not continuous at x 1 but differentiable at x 1 B f is differentiable at all x D 4 C f is continuous and differentiable at x 0 D f is continuous but not differentiable at x 1 E None of these Which of the following is true exists find the value of k E 8 D 4 3 B 8 C 4 3 8 10 2 8 8 4 874 34 40 8 8 E 2 3 2 10 2 4 2 4 2 4 2 4

AP Multiple Choice Problems 5 Derivatives and Limits No Calculator Be sure to show some work or give an explanation A 1 only B 2 only C 4 only D 1 and 4 E 2 and 6 2 f x does not exist for x Use the graph to the right for questions 1 3 The graph of f x consists of two line segments and a semicircle 1 f x 0 for x A 1 only 3 f 3 X f x B 2 only A 8 8 1 0 1 2 8 10 C 1 and 2 B 12 C 1 1 4 1 6 14 22 0 C 18 1 Name Date D 2 D 2 and 6 4 A differentiable function f has the values shown in the table below Estimate f 1 3 9 13 23 E 1 2 and 6 E 3 4 5

the function defined above then lim f x is B 0 f x x 1 2 for a for a 1 1

2 4 3 1 5 2 O 3 1 5 0 2 O 2 3 1 5 Find the solutions to the function 2 O 1 5 1 3 5 4

Which function has a domain of 0 2 U 2 00 O 3x x 2 20 x 2 O x 2 8

What are the extrema of f x 2x 2 4x 2 What are the domain and range of the function

8 A wedding caterer charges a 200 service fee and 24 per plate cost of food for one guest Johanna has gotten RSVPs from 72 guests but is still waiting to hear back from the other 18 She als has to pay for her own food and the groom s food who are not considered guests Let C p represent Johanna s catering cost for p plates of food a What is the domain of C p b What is the range of C p

2 Sketch a graph that satisfies the given conditions lim f x 2 x 3 lim f x 4 x 3 lim f x 2 x 12 f 3 3 f 2 1

Find the average rate of change of g x 4x 5 from x 3 to x 4

If x u v X then the line x is said to join of u and v The points u and v are said to adjacent if x u v We say that the points and the line x are incident with each other If two distinct lines x and y are incident with a common point then they are called adjacent lines A graph with p points and q lines is called a p q graph Note When there is no possibility of confusion we write V G V and X G X Examples Let V a b c d and X a b a c a d G V X is a 4 3 graph This graph can be represented by a diagram as shown in Fig 1 1 ob 1 Fig 1 1 In this graph the points a and b are adjacent whereas b and c are non adjacent

A point v of degree 0 is called an isolated point A point v of degree one is called an end point Note Loops are counted twice Example Consider the following 4 5 graph Fig 1 13 1 3 Fig 1 13 d 1 2 d 2 4 d 3 2 d 4 2 Total degrees 10 2 x 5 Theorem 1 1 The sum of the degrees of the points of a graph G is twice the number of lines in G i e Ed vi 2q Proof Every line of G is incident with two points Every line contributes two degrees There are q lines in p q graph 1d v 2q 2x number of lines in G Theorem 1 2 In any group G the number of points of odd degree is even Proof Let V V2 V denote the points of odd degree and w W Wm denote the points of even degree in G By theorem 1 1 1 d v 1 d w 2 q which is even

A wedding caterer charges a 200 service fee and 24 per plate cost of food for one guest Johanna has gotten RSVPs from 72 guests but is still waiting to hear back from the other 18 She also has to pay for her own food and the groom s food who are not considered guests Let C p represent Johanna s catering cost for p plates of food a What is the domain of C p b What is the range of C p

Math 41 oblem 4 The function W h 25 h 800 7 7 inches tall where 62 h 76 approximates the recommended minimum weight a Find W 70 and interpret the meaning of W 70 include units b Find the inverse function h W Show all your steps c In words explain what the inverse function calculates d Use your answer to part c to evaluate h 150 and Interpret its meaning include units e Graph the functions W h and h W on separate axes and label all axes owned by a company depreciates in value over time 2400t 10000

roblem 2 Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the following reduced row echelon form Solve the system 0 0 0 6 0 0 3 0 1 0 3 0 0 1 5 0 0 A x 6 35 9t B x 9 6s 3t C x 6 3s 9t D no solution x2 S inconsistent x3 9 3t X4 4 5t x5 1 x2 5 x3 3s 91 X4 5s 4t x5 1 9 4 00 0 F x 9 0 x3 9 X4 4 X5 0 G x 9 6s 3t X2 S x3 9 31 X4 4 5t X5 1 x2 S x3 3s 9t X4 5s 4t X5 t E x 6 3s 9t X2 5 3s 9t X4 5s 4t x5 1

Select the FIRST correct reason why the given series converges NOTE Make sure that you can provide the proper arguments to support your answers A Convergent geometric series B Convergent p series C Integral test D Comparison with a convergent p series E Converges by limit comparison test F Converges by alternating series test n n 1 1 1 n 8 n 2 n4 8 sin 2n 5 6 n 2 n 1 n cos NT In 3n 3 n ln n

Consider the series 1 an where 8 5n 6 5n 5 In this problem you must attempt to use the Ratio Test to decide whether the series converges Compute an an 1 L lim n 8 an Enter the numerical value of the limit L if it converges INF if it diverges to infinity MINF if it diverges to negative infinity or DIV if it diverges but not to infinity or negative infinity L Which of the following statements is true A The Ratio Test says that the series converges absolutely B The Ratio Test says that the series diverges C The Ratio Test says that the series converges conditionally D The Ratio Test is inconclusive but the series converges absolutely by another test or tests E The Ratio Test is inconclusive but the series diverges by another test or tests F The Ratio Test is inconclusive but the series converges conditionally by another test or tests Enter the letter for your choice here Activ Go to

256 a Find the point on the graph of y ex where the tangent line has slope 5 b Plot the graph of y ex and the tangent line whose slope is 5 a The point on the graph where the tangent line has slope 5 is Type an ordered pair Type an exact answer

Find In In

Use a graphing utility to approximate the real solutions Round to two decimal places x 3x 3x 6x 1 0 Solution 1 Solution 2

23 a lim x 4 x 5 x 4 b lim x 4 x 5 x 4 x 5 c lim x 4 x 4

21 44 Determining limits analytically Determine the following limits 21 a 1 lim x 2 x 2 1 x 2x 2 b lim 1 x 2x 2 c lim

sthenes estimated the circumference of the Earth as 250 000 stadia the length of an athletic stadium If we assume he used the most common length of stadia of his time his estimate of the circumference of the Earth would be 46 620 km Currently the accepted average circumference of the Earth is 40030 2 km Find the percentage error of Eratosthenes estimate of the circumference of the Earth a Calculate the actual and an approximate temperature in C using the standard formula and Tommaso s grandmother s rule b Calculate the percentage error of the approximate tempe vo

K Graph the function and then find the specified limits If necessary state that the limit does not exist f x 3x find lim f x and lim f x x 0 X 2 use the graphing tool to graph the function Click to enlarge graph Find lim f x Select the correct choice below and if necessary fill in the answer box to complete x 0 your choice OA lim f x x 0 B The limit does not exist Find lim f x Select the correct choice below and if necessary fill in the answer box to X 2 complete your choice OA lim f x X 2 B The limit does not exist C 10 8 4 2 10 18 Ay 6 4 2 2 4 6 8 10 2

Graph the function and then find the specified limits When necessary state that the limit does not exist 3x 10 for x 4 for x 4 F x OA X Find lim F x lim F x lim F x X 4 X 4 X 4 OA lim F x X 4 OB The limit does not exist OA OB Ay IM Find lim F x Select the correct choice below and if necessary fill in the answer box within your choice x 4 lim F x X 4 OB The limit does not exist O C Find lim F x Select the correct choice below and if necessary fill in the answer box within your choice X 4 Ay Q O D Q Time Remaining 01 23 11

Is the function given by f x 1 x 7 for x 3 4x 4 for x 3 Choose the correct answer below continuous at x 3 Why or why not OA The given function is not continuous at x 3 because f 3 lim f x 8 x 3 OB The given function is continuous at x 3 because f 3 lim f x 8 X 3 OC The given function is not continuous at x 3 because the limit does not exist D The given function is continuous at x 3 because f 3 does not exist

Use the graph of g to find the y a b lim g x x 0 1 lim g x e 40 2 c lim g x x 0 DNE d lim g x x 2 2 lim g x X 2 0 f lim g x x 2 g g 2 4 2 20 4 9 x