Limits & Continuity Questions and Answers

Consider the following functions f x 4x and g x x 3 Step 1 of 2 Find the formula for f g x and simplify your answer Then find the domain for f g x Round your answer to two decimal places if neces Answer f g x Keyboar

Evaluate lim O 8 3x 7 X 5 x 5 O 0 O Does not exist numerically

Evaluate lim 3x4 2x 4 numerically X O O O 3 04

Evaluate the following limit lim x 1 e4x 1 X 0 O O O lim x e4x 1 3 vnt V lim x e4x 1 1 vnt lim x e4x 1 0 nt V lim x e4x 1 4 vnt

Evaluate the following limit lim 5x 1 cotx x 0 O lim 5x 1 cotx e5 vint O lim 5x 1 cotx 1 vint O lim 5x 1 cotx vint O e lim 5x 1 cotx 5 nt

Approximate the area of the region bounded by y x 6 the x axis x 0 and x 4 by finding the combined area of the rectangles as shown in each figure and averaging the results 18 onts 4 14

Find the following indefinite integral 4 36x 18x 8 17 dx 9 O 16x5 9x 34x c 21 1 O20x5 6x 8x 34x c 1 Pre 3

Use the even odd properties of the trigonometric functions to find the exact value of the given expression Do not use a calculator cot T Select the correct choice below and fill in any answer boxes in your choice OA cot T Simplify your answer including any radicals Use integers or fractions for any numbers in the expression B The answer is undefined

Find the following limit or state that it does not exist 2 x 4x 3 X 1 lim X 1 Simplify the given limit 2 x 4x 3 lim X 1 X 1 O A lim X 1 lim Simplify your answer X 1 Evaluate the limit if possible Select the correct choice below and if necessary fill in th answer box to complete your choice x 4x 3 X 1 Type an exact answer OB The limit does not exist

Analyze the following limits and find the vertical asymptotes of f x a lim f x X 15 b lim f x X 15 C lim f x X 15 x 15 2 x 225 a Rewrite the given limit in a way that can be evaluated using direct substitution x 15 lim lim Simplify your answer 2 x 15 x 225 x 15 Now evaluate the limit Select the correct choice below and if necessary fill in the answer box to complete your choice lim f x OA Simplify your answer X 15 B The limit does not exist and is neither o nor co b Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x Simplify your answer x 15 B The limit does not exist and is neither co nor co c Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x Simplify your answer X 15 B The limit does not exist and is neither nor 00 Where are the vertical asymptotes of the function if any exist Select the correct choice below

Find the following limit or state that it does not exist lim h 0 49 h 7 h Simplify the given limit 49 h 7 h lim h 0 lim OA lim h 0 h 0 Evaluate the limit if possible Select the correct choice below and if necessary fill in the answer box to complete your ch Simplify your answer 49 h 7 h Type an exact answer OB The limit does not exist

Evaluate the following limit Use l H pital s Rule when it is convenient and applicable lim 818 lim x In 2 8x 3x In 10 X 10 X 11 Type an exact answer

Use the graph of g in the figure to find the following values if they exist If a limit does not exist explain why a g 2 b lim g x c lim g x e g 3 d lim g x x 2 g lim g x h g 4 X 3 f lim g x X 3 i lim g x a Select the correct ansiver below and if necessary fill in the answer box within your choice OA g 2 Type an integer or a decimal B The value g 2 does not exist because g is not defined at x 2 b Select the correct answer below and if necessary fill in the answer box within your choice accos OA lim g x Type an integer or a decimal X 2 B The limit does not exist because g is not defined for all x sufficiently close to 2 with x 2 c Select the correct answer below and if necessary fill in the answer box within your choice OA lim g x Type an integer or a decimal X 2 B The limit does not exist because g is not defined for all x sufficiently close to 2 with x 2 2 ByLoc

The following graph shows the average price of a movie ticket for the years 2009 and 2010 Complete parts a and b below O C a Draw a bar graph that shows the entire scale from 0 to 8 Choose the correct graph below O A Average Price of a Movie Ticket OB Q 2009 7 01 2010 1 012 1 2 2009 2010 0 3 2 4 5 6 Price Average Price of a Movie Ticket 7 63 7 01 3 7 63 4 5 7 8 6 7 Tx 8 Q S O D 2009 2010 2009 7 01 2010 Average Price of a Movie Theat 0 1 2009 2010 7 0 Average Price of a Movie Ticket 72 74 2 7 01 7 63 4 3 5 6 Price Average Price of a Movie Ticket 7 63 7 01 7 63 7 6 7 8 1 2 3 4 7 5 6 7 85 8 r

Find the supplementary angle of 158 6 The supplementary angle of 158 6 is Type an integer or a decimal

Solve each equation x 5 log4 256 Ox 9 O x 1 O x 1 O x 4 Question 16

Evaluate each expression 1 1082 16 32 0 4 O 0 41 1 8

Evaluate each expression 1 1087 49 O 29x10 1 0 57 0 2

Write each equation in logarithmic form 7 1 O log 0 1 O log 1 0 O log 7 0

Siven f x find g x and h x such that f x g h x and neither g x nor h x is solely x f x x 4 3 nswer g x

For the function graphed below find the values in the right column 5 0 0 4 4 2 4 4 10 A limf x B lim f x C lim f x D lim f x x 4 E lim f x F f 4

Use this information for questions 18 and 19 You will need to find a polar equation with a focus at the pole and given eccentricity and directrix e 1 x 5

olve the following triangle using either the Law of Sines or the Law of Cosines B 30 C 87 b 6 elect the correct choice below and if necessary fill in the answer boxes to complete your choice Round side lengths to the nearest hundredth and angle measures to the nearest degree as n OA There is only one possible solution for the triangle The measurements for the remaining angle A and sides a and c are as follows A C a OB There are two possible solutions for the triangle The measurements for the solution with the smaller angle A are as follows A a C The measurements for the solution with the larger angle A are as follows A a C C There are no possible solutions for this triangle

Solve the triangle Round to one decimal place as needed 4 B

Question 3 Consider the graph of some function y f x 10 Evaluate lim f x lim f x and lim f x X 2 X 1 X 4 lim f x 3 X 2 lim f x 4 X 1 lim f x 3 lim f x 3 X 2 lim f x does not exist X 1 lim f x does not exist lim f x 5 X 2 lim f x does not exist X 1 3 lim f x 3 X 2 O lim f x does not exist X 1 5 lim f x 3 VIA 10 0 5 10 OMark this question 10

Let f x x 2x on 5 1 Use the IVT to determine if there is a solution to f x 15 in the interval between 5 and 1 If so find the value of c in the interval such that f c 15 O O O O f x is continuous on 5 1 15 is not between 5 and 1 so the IVT does not apply f x is continuous on 5 1 15 is between f 5 35 and f 1 1 c 5 f x is continuous on 5 1 15 is between f 5 35 and f 1 1 c 3 c 5 f x is continuous on 5 1 15 is between f 5 35 and f 1 1 C 3

Use a table of values to evaluate lim Olim 2x 250 x 5 x 25 Olim 2x 250 x 5 x 25 2x 250 Olim x 5 x 25 2x 250 O lim x 5 x 25 15 8 15 00150 does not exist 2x 250 x 5 x 25

9x 5 Consider the function f x x 7 Evaluate lim f x and lim f x x 2 X 2 O O O lim f x 13 x 2 lim f x 11 vint lim f x 3 x 2 lim f x 13 v424 lim f x 13 X 2 lim f x 3 v 24 lim f x 4 X 2 lim f x 2 v 24 if x 2 if x 2

Suppose 3x 24x 53 f x 3x 24x 43 for all x values near 4 except possibly at 4 Evaluate lim f x x 4 Olim f x does not exist 415 Olim f x 5 1 lim f x 91 415 O lim f x 5

Consider the following function f x x 1 if x 3 2x 22 if x 3 Determine if f x is continuous at x 3 If not select the option with the correct reasoning as to why not O Not continuous at x 3 because lim f x does not exist O Not continuous at x 3 because lim f x f 3 O Not continuous at x 3 because f 3 is undefined O Continuous at x 3

Consider the graph of some function y f x Evaluate lim f x lim f x and lim f x X 2 x 1 X 4 O 10 lim f x 3 X 2 Olim f x 4 x 1 lim f x 3 lim f x 3 x 2 lim f x does not exist lim x does not exist lim x 5 Olim f x does not exist 1 lim x 3 A lim x 3 X 2 Olim f x does not exist X 1 5 dia 10 0 5 10 10

For 0 find 0 necessary to prove lim 3x 20 26 X 2 0 0 11 30 O O 5 4 0 0 2 O 0 0 1 10 O 3

Determine the area of the triangle A Type a cm m cm 80 cm 4 m

Use the factor theorem to check whether the polynomial x 2 is a factor of the polynomial 2x 8x 16x 19 Find the remainder when the polynomial 2x 8x 16x 19 is divide polynomial x 2 Is the polynomial x 2 a factor of the polynomial 2x 8x 16x 19 Yes No IX What is the remainder when the polynomial 2x3 8x 16x 19 is divided by the polynomial x 2 Simplify your answer

e cost of controlling emissions at a firm is given by C q 7000 120 q where g is the reduction in emissions in pounds of pollutants per day and C is the daily cost to the firm in dollars of this reduction Government clear air subsidies mount to 840 per pound of pollutant removed How many pounds of pollutant should the firm remove each day in order to minimize net cost cost minus subsidy

6x 12 x 7 The graph of y has a vertical asymptote What is it

Evaluate lim 10 cos 6x 7x e 1 Give an exact answer

8 7 5 4 3 2 2 1 Find lim 1 2 3 4 6 7 8 1 2 3 4 5 6 7 Function g x is graphed above 5 g x 20 6 4x 24 C

used car salesman earns 2 9 on the sale of a car that sells for 59 890 Calculate the amount of commission he would make

The rate of markup on a bicycle selling for 845 is 28 Calculate the cost

retailer buys a handbag for 57 80 He marks it up by 55 of his cost Calculate the selling price

Smartphones are selling at 550 each and are marked up 133 above cost Calculate the rate of margin

Lumber is selling at 45 each and are marked up 25 above cost due to demand Calculate the rate of markup

Video games are selling at 48 each and are marked up 6 80 above cost What is the cost

Which of the following is a function x y 81 O 5y 10x 35 O 5x 4y 20 y 7 9x

Consider the following piecewise function f x O Evaluate f 5 f 3 and 2 O O 3x 7x 11 x 7x 4 O f 5 29 f 3 5 f 2 16 f 5 51 f 3 59 f 2 18 f 5 246 3 113 F 2 18 x 3 X 3 f 5 29 f 3 5 f 3 18

Consider the function f x 3x 5x 2 Use It to find 4 f 1 and f a 3 f 4 126 0 1 3 f a 3 90 50 14 f 4 30 O f 1 6 fln 3 30 130 14 4 30 O f 1 9 f a 3 90 50 14 f 4 30 O 1 10

3 2 points Find the limit if it exists lim 1023 3 7 5x3 818

What number shoud we write inside if we want to have lim 4x6 x 2x x 2x 2 x118 Type your step by step solution in the space provided

Suppose 61 ax 41 5 x x ar 12 for x 2 and x near 2 Determine a in such a way that the Squeeze Theorem can be used to find lim g x Then find this limit Type your detailed solution along with explanation for x 2 what you did