Limits & Continuity Questions and Answers

10 The function f x is known to be continuous determine the value of a and the value of b required to make this statement true T4 6 x x 1 f x ax b 1 x 2 5x 3x 2

9 Determine the limit of lim x 8 2 X x 8 T4

5 Determine if the limit exists at x 1 and if so determine the value 2x 3 x 1 f x 6 x 1 3x 2 x 1 K2

4 What is limx 4 x 2x 3 a 3 b 2 c 3 d 21

As x f x As x 5 f x As x 5 from the right f x lim f x x 3 lim f x X 6 graph of f x below 6 What is the equation of the horizontal asymptote What is the equation of the vertical asymptote Classify the discontinuity at x 3 Classify the discontinuity at x 6

f x I 2 kx x 1 5 3x x 1 207 If lim f x 3 and lim g x 7 find lim x 2 x 2 x2 f x 208 Find a value for k so that f x is continuous if g x

206 Find lim 72 3

101 Solve the inequality 1 1 4 x 2 0

109 Find the average rate of change between x 2 and x 5 based on the graph below 11 10 9 6 54321 Ay y f x I 4 5 6 7 8 9 10

Apply L H pital s Rule to evaluate the following limit It may be necessary to apply it more than once 5x 5 x 7 2 lim

4 No calc The x coordinate of the point where the line tangent to the parabola y 8x at 2 4 intersects the axis of symmetry of the parabola is x A 1 B 1 C 2 D 0 5 No calc If y 3sin x 4 cos x then y y A 6sin x 8 cos x B 6sin x 8 cos x C 6sinx 8cos x D 6sin x 8 cos x E 0 6 Calc Let f be the function given by f x tan x and let g be the function given by g x x At what value of x in the interval 0 x do the graphs off and g have parallel tangent lines A 0 B 0 660 7 No calc The lim A40 A 1 C 2 083 h 8 No cale lim A 4 D 2 194 at x 3 is B 0 1 D 3 lim 1370 13 0 h 70 x 16x 20x 4x B 2 I m 0 0 0 0 E 2 5 D 2 E 2 207 E DNE E 4

3 Use the graphs of f and g to find the domain of the function indicated a f g b f g c fog 2 f x 4 Figure 1 Graph of f x and g x g x

1 Calculate each of the following and simplify your answer f g fg fog go f a f x x and g x 1 b f x x and g x x 1 X x x 1

104 If lim f x 4 and lim g x 5 find lim 2f x g x

figure shows the graph of a quadratic function y f x AY 20 y f x 16 12 8 16 4 12 4 8 12 16 Q How many real solutions are there f x 1 Choose the correct answer below OA One OB Two OC None

y x 1 1 The whether its graph opens upward or downw Select the correct choice below and fill in the answer box to complete your choice Type an ordered pair OA The vertex of the parabola is OB The vertex of the parabola is and it opens downward and it opens upward

e quadratic function defined a write the function in m P x a x h k b give the vertex of rabola and c graph the function x 2x 7 plify your answer Use integers or fractions for any bers in the expression 516 7 512

have real solutions tell whether they are rational or irrational Do not actually solve the equation 9x 18x 9 0 VYRIGHT OF Tiny is TO CUTTOULOTTUVIL VI I qyuuurucu vyuuuvIICIA b 4ac b b 4ac b b 4ac b 4ac The discriminant is 0 Simplify your answer Type an integer or a simplified fraction Determine the number of real solutions of the equation If the equation does have real solutions tell whether they are rational or irrational Select the correct choice below and if necessary fill in the answer box to complete your choice OA The equation has B The equation has OC The equation has no real solutions UAL UI real rational solution s Type a whole number real irrational solution s Type a whole number

4 Given this graph of the function f x Find a f 4 e x when f x 2 b f 0 1 x f x when f x 0 c f 3 d f 5

The graph of f is given State with reasons the num bers at which f is not differentiable 35 2 0 fr X 4

5 Graphs of the velocity functions of two particles are shown where t is measured in seconds When is each particle speeding up When is it slowing down Explain a b VA VA 0 1 0 1

Come up with an algebraic expression for a function with exactly two discontinuities It can contain familiar functions and it can be a piecewise function Your task will be to find the discontinuities in another team s function Answer

f x x 3 2 x 3 x 3 x 3 1 Evaluate f 3 in the piecewise function above Type a response

1 Use the graph of g x shown to the right to answer the following questions A lim 9 x B lim g x C lim g x 111 D miT 10touder I sio C libero om 6 A omak 1 3 2 1 y g x 1 2 g 1 Hoy of inpiroqni ai phigini la tag of vow to Host 3 4

x 9 9 Given f x 7 3x A Find lim x 818 B Find lim f x 0 C Find lim f x 2 0

3x xoo x 5x 10 Find lim rose etning 2 e x 1 0 e 1 11 Find lim

B

olve the A log6x 2 x

Evaluate the integral using integration by parts Sen xe x dx 6x dx

7 lim x 0 Hint Use the identity Vx x x 0

2x 2 17 Finding a symmetric interval Let f x and note x 1 that lim f x 4 For each value of e use a graphing utility to find all values of 8 0 such that f x 4 whenever 0 x 1 d a 2 b 8 1 IC c For any 0 make a conjecture about the value of 8 that sat isfies the preceding inequality

1 lim 2x 6 Hint Use the inequality a b a bl M x 3 which holds for all constants a and b see Exercise 74

1 lim x 16 x 4 x 4 8 Hint Factor and simplify

K A news item is spread by word of mouth to a potential audience of 6 000 people The following function models the number of people who have heard the news after t days Complete parts a through g below 6 000 1 70e 0 3t f t a Approximately how many people will have heard the news after 12 days 2100 Round to the nearest hundred as needed b At approximately what rate will the news spread after 18 days 300 people day Round to the nearest hundred as needed c Approximately when will 2500 people have heard the news 2500 people will have heard the news after approximately Round to the nearest whole number as needed days Ay 6000 4000 2000 0 0 5 10 15 20 25 30 graph of f Q G Ay 500 300 100 oo 0 5 10 15 20 25 30 graph of f E

The angle of elevation to the top of a very tall Building is found to be 8 from the ground at a distance of 1 mi from the base of the building Using this information find the height of the building Round your answer to the nearest foot ft

The bar graph on the right shows the average cost of room and board at four year public and private colleges for four selected years from 2011 through 2017 The data can be modeled by f x 945x 3143 and g x 3851e 0 131x in which f x and g x represent the average cost of room and board at public four year colleges in the school year ending x years after 2010 Use these functions to complete parts a through c below ACCEDI Average Cost Average Cost of Room and Board at Four Year Colleges 16000 14000 Public Colleges 10 921 12000 Private Colleges 10 610 10000 8000 4973 5317 600046095317 a According to the linear model what was the average cost of room and board at public four year colleges for the school year ending in 2015 The average cost in 2015 was Round to the nearest dollar as needed 4000 2000 0 7927 5866 7596 2011 2013 2015 2017 Ending Year in School Year

Two tugboats that are a 125 ft apart pull a barge as shown If the length of one cable is b 208 ft and the length of the other is c 245 ft find the angle formed by the two cables Round your answer to the neares degree poboboboto b

37 50 Horizontal asymptotes Determine lim f x and lim f x x18 8116 for the following functions Then give the horizontal asymptotes of f if any 37 f x 39 f x 41 f x 43 f x 45 f x 47 f x 49 f x 4x 20x 1 6x 9x 8 3x 2 3x 7 x4 5x2 40x5 x 16x4 2x 1 2x4 4x89x4 4x 1 2x 16x6 1 x6 8 4x 3x4 1 38 f x 40 f x 42 f x 44 f x 46 f x 3x 7 x2 5x 12x 3 3x8 2x7 2x 1 3x4 2 6x 1 4x4 3x 1 x 1 2x 1 48 f x x x 9x 50 f x 4x 3x 9x 1

11 37 50 Horizontal asymptotes Determine lim f x and lim f x for the following functions Then give the horizontal asymptotes of f if any 37 f x 39 f x 41 f x 45 f x 47 f x 4x 20x 1 6x 9x 8 3x 2 40x x 43 f x 16x4 2x 3x 7 x4 5x 38 f x 1 2x4 4x8 9x4 4x 1 2x 16x6 1 40 f x 42 f x 44 f x 46 f x 3x 7 x 5x 12x 3 3x8 2x7 2x 1 3x4 2 6x 1 4x4 3x 1 x 1 2x 1 48 f x x x 9x

13 lim e lim e and lim et 18 8118 118

20 Fifty part time students were asked how many courses they were taking this term The incomplete results are shown in the table below of Courses Frequency 30 15 1 2 3 Relative Frequency Cumulative Relative Frequency 0 6 a Fill in the blanks in the table b What percent of students take exactly two courses c What percent of students take one or two courses d Describe the difference between the Relative Frequency and the Cumulative Relative Frequency columns

The function f x 10x ln 1 x is represented as a power series Find the following coefficients in the power series C2 C3 CA C5 Find the radius of convergence R of the series 8 f x Cnz 7 0

3 For each of the following functions find all values a in o at which the function is discontinuous if there are any At each such value compute the two one sided limits lim f x and x a In x lim f x x and say what type of discontinuity removable discontinuity jump discontinuity vertical asymptote or other occurs at a If there is a removable discontinuity explain how you could redefine the function at that point to make it continuous a h x b g t ln let 4 c f x COS d g x arctan

ing calculator is recommended ow is the number e defined e is the number such that lim De is the number such that lim Oe is the number such that lim h 00 Oe is the number such that lim h 0 e is the number such that lim im 40 h ho h lim 2 8 1 0 h 2 7 h 1 h 0 99 e h 1 1 1 1 eh eh e h 1 1 h se a calculator to estimate the values of the limits correct to two decima 2 7 1 X 1 1 h X 1 h 0 h 1 1 hat can you conclude about the value of e 0 99 e 1 03 0 99 e 2 7 1 03 e 2 7 2 7 e 2 8 oces1

g x lim x 3x X 3 3 3 3 0 1 3 5x x 1 x 3x 1 x 3 2x 9 x 3 8 The piecewise defined function g is given above Find the following values if possible Show all work a lim g x 3 b lim g x c g 3 x 3 lim xo 35x 3 5 1 3 5 4 cr 2 x 3x 3 2 9 0 L

Find the radius of convergence of the power series Your answer should be a nonnegative real number or INF Radius of convergence is ENG AMBIENT ABRIELE Dements Merancie www BOTH D Pros Entlast MONTH RACHELLES orien naniganty ensarent BALONES AURE ww OBENIAME THREA HARGE MARDARY WEB 22 1 3 26 26

Investment A is currently worth 80 500 and is growing at the rate of 13 per year compounded continuously Investment B is currently worth 70 000 and is growing at the rate of 14 per y compounded continuously After how many years will the two investments have the same value The investments will have the same value after years Do not round until the final answer Then round to the nearest tenth as needed

Theorem 1 If lim f x L and lim f x M then L M x p x p Proof Suppose LM then I L MI 0 Since lim f x L if we take x p then 38 0 such that Ix pl 8 1 f x LI e Similarly since lim f x M 38 0 such that x p Ix pl 8 lf x MI E If we choose 8 min 8 8 then Ix pl 8 will mean that Ix pl 8 and lx pl 8 In this case we will have both I f x Li e as well as I f x MI E So that IL MI IL f x f x MISI f x LI f x MI That is we get L MI IL MI which is a contradiction Therefore our supposition is wrong Hence L M We now state and prove a theorem whose usefulness will be clear to you in Unit 4 ii lim f x L lim h x X 4 Xa Theorem 2 Let f g and h be functions defined on an interval I containing a except possibly at a Suppose i f x g x Sh x x I a Then lim g x exists and is equal to L x a Proof By the definition of limit given 0 38 0 and 8 0 such that If x LI e for 0 ix al 8 and Ih x LI e for 0 lx al 8 Let 8 min 8 8 Then 0 lx al 8 If x LI e and h x LI E L ESf x SL E and L E h x L E We also have f x g x Sh x x E I a Thus we get 0 1x al 8 L es f x g x Sh x L In other words 0 lx al 8 lg x LI E Therefore lim g x L 2 IL MI Theorem 2 is also called the sandwich theorem or the squeeze theorem because g is being sandwiched between f and h Let us see how this theorem can be used IL MI 2 Example 4 Given that I f x 11 3 x 1 2 xe R can we calculate lim f x x 1 We know that 3 x 1 f x 1 3 x 1 2x This means that 3 x 1 1 f x 3 x 1 1 lim 3 x 1 1 1 lim x Using the sandwich theorem and the fact that 3 x 1 1 we get lim f x 1 31 1 X 1 x 11 In the next section we will look at the limits of the sum product and quotient of functions Theorem 3 Let f and g be two functions such that lim f x and lim g x exist Then x p x p i 2 2 1 Algebra of limits Now that you are familiar with limits Let us state some basic properties of limits Their proofs are beyond the scope of this course ii lim f x g x lim f x lim g x x p x p x p lim f x g x lim f x im 8 x x p Sum rule Product rule

Find the point closest to the origin in the line of intersection of the planes y 2z 10 and x y 11 The point closest to the origin is Type an ordered triple

0 13 Points DETAILS A graphing calculator is recommended a How is the number e defined SCALCET9 Oe is the number such that lim h 0 Oe is the number such that lim h 1 O e is the number such that lim h00 lim h 0 lim h 0 eh 1 1 h h e O e is the number such that lime h 0 O e is the number such that lim 1 1 h eh 1 1 h e h 1 1 h b Use a calculator to estimate the values of the limits correct to two decimal place 2 7h 1 h 2 8h 1 h 1 1 h What can you conclude about the value of e O 0 99 e 1 03 O 0 99 e 2 7 O 1 03 e 2 7 O 2 7 e 2 8 0 0 e 1