Limits & Continuity Questions and Answers

Use algebra and the properties of limits as needed to find the given limit If the limit does not exist say so lim 8x 3 Select the correct choice below and if necessary fill in the answer box to complete your choice A lim 8x 3 Type an exact answer using radicals as needed X 4 OB The limit does not exist

Which logarithm is equal to x log 4 Select the correct answer below O log 4 x O log x O log 4 O log 4 x O log O log

Evaluate the definite integral 1 5x 4x 7 dx 0 1 5x 4x 0 5x 4x 7 dx Simplif

Use the properties of limits to evaluate the limit if it exists 3x 7x 2 x 13x 4x 2 lim OA OB 0 O C C 8 3 7 4 Does not exist

Use the properties of limits to find the given limit 5x5 2x4 10x 20 6x5 lim X lim 848 5x5 2x4 10x 20 6x Type an integer or a simplified fraction

Solve the triangle Round lengths to the nearest tenth and angle measures to the nearest degree B 9 5 A

Solve the equation on the interval 0 2TT sin x cos X 0

1 Below is the graph of y 4 Translate it to become the graph of y 3 X S

f x Find f x for each x value in the table X 3 2 1 0 f x 0 0 8 08 3 X

When asked for exact values you MUST draw each angle on the Cartesian Plane identify the related acute angle and drav the special triangle or use the unit circle Determine the exact value for the expression using a compound angle formula 5 5 12 12 COS T COS 4 sin T 4 sin compound angle formula cos A B COSA COSB Cos TT cos 577 12 sinA sin B sin sin 5 12 A 4 B 57 12 Cos 4 5 12 cos 3

The fun V g x 8 Find g x for each x value in the table X 3 2 1 g is defined by the following rule 0 g x 0 0 0 3 08 X

9 x 13 Find g x for each x value in the table X 2 1 0 g x 8 08 X 5

Use the Factor Theorem to find all real zeros for the given polynomial function and one factor f x 5x 11x 3x 10 x 2 What zero corresponds to the factor x 2 H What are the other real zeros Select all correct answers I H U I 1 101 5 0x 10 I 1 102 10 x 1 101 1 101 1 102 10 1 101 10 There are no other real zeros

Step 1 Isolate one radical on one side of the equation Step 2 Raise each side of the equation to a power equal to the index of the radical and simplify Step 3 If the equation still contains a radical term repeat Steps 1 and 2 If not solve the equatic Step 4 Check all proposed solutions in the original equation Caution a b a bn 4 Solve each equation for y a Solve y 2 1 x

The circles are tangent to the rectangle and to each other as shown If the area of each circle is 16 then the area of the rectangle is F 32 G 48 H 96 J 128 K 512 A D 167 16T B C

Determine the following limit Use L Hopital s Rule if appropriate Enter DNE if the limit fails to exist except in case of an infinite limit If an infinite limit exists enter or o as appropriate sin 3x lim sin 67

The graph of f is given below Sketch the graph of f 10 8 642 Clear All Draw 2 4 6 8 10 10 8 6 4 2 2 4 200 6 8 16

If tan 0 7 0 0 2 find sin DIN sin 2 0 2 Simplify your answer including any radicals Use inte

From the graph determine the x and y intercepts and the vertical and horizontal asymptotes If an answer does not exist enter DNE Enter your asymptotes as a comma separated list of equatio necessary x intercept x intercept y intercept vertical asymptote s horizontal asymptote x y 00 6 4 N smaller x value larger x value X

The drive wheel of an engine is 13 inches in diameter and the pulley on the rotary pump is 7 inches in diameter If the shafts of the drive wheel and the pulley are 2 feet apart and the belt is crossed as shown in the figure what length of belt is required to join them 6 5 in The length of the belt is approximately in Round the final answer to two decimal places as needed Round all intermediate values to four decimal places as needed 2 ft 3 5

Given A 4 1 43 4 1 and B 0 4 find the product 0 4 BA if possible 1 3

7 Rewrite the following geometric series into the form arn 1 Then state the initial term a the ratio r and determine whether they converge or diverge If it converges find n 1 the value the series converges to a Le n 1 2n n 2 b 1 n 1 n 1 4 1 3 n 1 8 d 2 4n 1 n 1

Rewrite equation as requested a Rewrite as an exponential equation a 1 1083 81 b Rewrite as a logarithmic equation 8 64 OP 0 lag 4 0 0 X 00

Tutorial Exercise Evaluate the definite integral Use a graphing utility to verify your result 1 Step 1 9 sin x dx To evaluate the integral M Therefore To 9 sin x dx first use the property of definite integrals that is S f x g x dx To Fox a 9 sin x dx f x dx 1 90 TO g x dx dx R sin x dx

Find the general solution of the differential equation Use C for the constant of integration 5x V1 x y 1 dy dx

By definition fog x So if g 2 6 and f 6 20 then fog 2

5 Compute the first four partial sums S S2 S3 and S4 for the series whose nth term 1 In other words compute k 72 In n 1 n for k 1 2 3 4

Determine the area of the given region under the curve 1 1 y

ind the area of the shaded region AY 2 1 1 4 y x 2 2 3 4 square units y x

John decided to get some exercise yesterday using a local hiking trail After driving to the local trail John decided to track his distance from the car parked in the trail parking lot and time Suppose John started at 8 am The graph shown below represents John s distance d in miles from his car t hours after 8 am 5 Distance mi 4 3 2 1 0 1 2 d t 3 Time hr 4 5 6 Using the image above suppose g t d t 3 4 Draw a picture of g t To complete this question draw a picture of the function take a picture of it and upload the resulting image to this question

6 Consider the infinite series 1 1 n 1 a Compute the first few partial sums of this series Do you see a pattern What does t sequence of partial sums look like b Using your sequence from part a determine whether the infinite series converges diverges

8 Determine whether the following series diverge using the divergence test If the test inconclusive please state it as such a b c n 1 n 1 n n 2 n n5 2n In n n Use L Hopital s

1 Let an and bn be two sequences with lim an 1 and lim bn 1 Use the algebraic limit laws to determine the following 81x 81x a lim 3an n x 4bn 1 1 b lim bn an n 2 2 c lim 7000an 1510bn n x

Use L H pital s rule to find the limit x 51 X 6 lim X X Apply L H pital s rule so that the lim 8 18 lim X 2 51 X 6 OA lim X 18 x Select the correct choice below and if necessary fill in the answer box to complete your choice 51 X 6 x 51 X 6 lim X 8 Type an exact answer in simplified form OB The limit does not exist is not an indeterminate form of the type co 00

Even though the limit can be found using the theorem Limits of Rational Functions at Infinity and Horizontal Asymptotes of Rational Functions use L H pital s rule to find the limit lim X 8 2x 6 5x 3 Apply L H pital s rule so that the lim X 8 2x 6 5x 3 O A lim X 8 Select the correct choice below and if necessary fill in the answer box to complete your choice lim X 8 2x 6 5x 3 lim X Simplify your answer OB The limit does not exist 2x 6 5x 3 WIE is not an indeterminate form of the type o 4

0 0 O 0 1 0 1 0 Evaluate Assume u 0 when In u appears In x X 12 1 0 OA O B dx O D In x 4 4x In x 4 4 1 0 O C Inx C C In x 2 C C

Use L H pital s rule to find the following limit and check the answer using an algebraic simplification x x 72 X 9 lim X 9 lim X 9 x x 72 X 9 Simplify your answer

7 0 Find the equilibrium point 1 0 7 0 1 0 07 1 11 0 16 S x x x 07 1 11 0 D x O A 16 16 B 16 4 07 1 11 0 OC 4 4 OD 4 16

Use the properties of summation and Theorem 4 2 to evaluate the sum 57 3 181

bwork2 colu 1 point Using the graph of g z below answer the following questions a Where is g x 0 b Where is g x 0 c Where is g x 0 d Where is g x 0 e Where is g x 0 1 Where is g x 0 NOTE When using interval notation in WeBWork remember that Enter an interval VA 1 0 Enter an interval Enter a list of values separated by commas Enter an interval Enter an interval Enter a list of values separated by commas

The graph of y f x is shown to the right Identify the intervals on which f x is increasing Which of the following shows every interval on which f x is increasing Choose the correct answer below O A b c e f g h OC a c d f O B a c e f g h D b c d f g h a b C d e I f g h G

07 0 11 0 07 0 11 0 07 0 11 0 07 0 11 0 Use the graph of y e and transformations to sketch the exponential function f x ex 6 Determine the domain and range Also determine the y intercept and find the equation of the horizontal asymptote 07 0 11 0 P 07 0 11 0 graph What is the domain of f x ex 6 07 0 11 0 Type your answer in interval notation Type your answer in interval notation What is the range of f x ex 6 07 2 What is the y intercept of f x ex 6 11 0 Simplify your answer Type an exact answer in terms of e What is the horizontal asymptote of f x ex 6 Type an equation 10 10 8 6 2 4 6 8 Ay

11 Given a triangle ABC let y 90 60 and a 5 Find the exact values of the remaining parts 4 marks G

The graph of y f x is shown in red Graph y f x 5 Choose the correct graph in blue on the right OA OC Q oo O B O D x Q G O

Na Du La Cu At D x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units D x e x 5 S x ex 7 a Find the equilibrium point b Find the consumer surplus at the equilibrium point c Find the producer surplus at the equilibrium point a What are the coordinates of the equilibrium point Type an ordered pair Round each coordinate to the nearest hundredth as needed Do not include the symbol in your answer

Rewrite the complex number 7 cos1 isin1 7 cos1 isin1 in a bia bi form Write the values in exact form or write out as many decimals as possible

Na a Find the equilibrium point b Find the consumer surplus and producer surplus Du c Suppose a maximum rent of 850 per month is imposed by the town council Find the point xc Pc La Next question Demand for apartments in a certain town is D x 1500 2x and the supply is S x 600 10x where x is the number of apartments in hundreds and D x and S x are the rent in dollars per month per apartment Cu At d Find the new consumer surplus and new producer surplus e Find the deadweight loss a Which of the following would be the best first step to find the equilibrium point O A Solve D x S x for x OB Find the value of q such that THE C Solve D x S x for x D Solve D x S x for x q 0 q D x dx S x dx 0

13 a D x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and c the producer surplus at the equilibrium point D x 6 x for 0 x 6 S x x 6 a What are the coordinates of the equilibrium point Type an ordered pair

Evaluate g x ln x at the indicated value of x x e 9 09 9 0 9

Na Ou La Cu At Next question D x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and c the producer surplus at the equilibrium point D x 6 x for 0 x 6 S x x 6 a What are the coordinates of the equilibrium point Type an ordered pair did