Limits & Continuity Questions and Answers

The graph of y f x solid and y g x dashed is shown 6 5 4 3 4 lim 2 1 9 x 0 8 f x lim 24 1 g z Does Not Exist 88 0 8 2 z 1 g z Does Not Exist 0 Find the following limits 1 Does Not Exist 6 5 4 3 2 1 2 3 nd 5 6 2 3 4 1 1 1 1 S 6 q

6 Here is a graph of a pattern of numbers where V is a function of n The first point is 0 1 22 20 18 16 14 12 10 8 6 28 26 24 4 2 32 30 V 2 3 4 IMMathematics 5 a Plot the next 2 points on the graph that follow the pattern b Write an equation to describe the relationship between V and n f 1 1 f n f n 1 3 n 41 3 8

The graph below is the function f x 8 7 6 5 4 3 2 8 7 6 S 4 lim 118 32 4 12 2 3 3 4 5 6 7 4 6 8 1 2 3 4 5 6 7 8 96 f x ter the limit If the limit is m enter go If the limit is enter oo If no limit exists

A cyclist rides down a long straight road at a velocity in m min given by v t 100 10t for 0 st 10 a How far does the cyclist travel in the first 4 min b How far does the cyclist travel in the first 8 min c How far has the cyclist traveled when his velocity is 75 m min

Check all that apply sin 5x sin CT 5 5x 5x sin 6x 5x 6x sin 5x sin 6x 6 sin 5x 6x

The graph of y f x solid and y g x dashed is shown AT 4 3 Find the following limits f x lim 2 4 g x 0 lim f x lim 2 4 g z 8 Does Not Exist 0 2 1 f x g z Does Not Exist 6 5 4 3 2 1 1 2 3 A 5 6 1 2 3 4 5 6

State whether the function is continuous at the given x values Justify your answers sqvt done lar 2x 1 x 1 7 f x 2 2 1 x 4 x 4 a x 3x at x 1 b at x 4 2 4 1 4 3 4 3 4

The graph of y f x solid and y g x dashed is shown 6 5 4 B Find the following limits f x lim 1 g x 818 0 Does Not Exist f x lim 2 1 g x 00 Does Not Exist 1 f z lim 2 1 g x 6 5 4 3 2 1 2 3 4 5 6 3 4 5 6

Find the limit sin 7x lim x0 sin 6x lim x 0 tan 6x sin 7x

To find lim 2 0 cos x 1 which of the following steps might be useful 8 Multiplying numerator and denominator by Adding 1 to numerator and denominator Multiplying numerator and denominator by cos x 1

Evaluate 3 2 3x lim 4x sin 2 0

The graphs of y f x solid and y g x dashed are shown The two curves intersect at 5 0 Determine the sign of the following limits You may assume the limits exist lim f x g x is 5 Positive Negative Zero lim f x g x is 20 Positive Negative Zoro

Find the value s of k such that f x is continuous everywhere Sx 76 a 2 4x3 3kx 2 x 2 q f x

x 3x 7 x 3 3 x 1 x 1 8 g x 2x 1 In x a at x 3 ran b at x

Find the limit sin 5x 3x lim x 0 3x 40 sin 5x lim

The graph below is the function f x 8 7 6 5 4 3 2 lim f x 2 8 8 7 6 5 4 3 27 1 2 AWN 3 4 567 7 8 4 5 6 7 Enter the limit If the limit is o enter oo If the limit is enter oo If no limit exists enter

Suppose f x has the following values I f x lim 8 7 9 7 99 6 5 6 05 1 6 f T 7 999 8 6 005 9 appears to be 8 001 6 004 8 01 6 04 8 6

Given lim f x 2 and lim g x 1 evaluate 8118 8178 lim g x ln 6f x 4 8118 lim 8118 lim 8118 lim 8118 69 x e f z 6 2 49 x cos f x a x C

Evaluate the limit 4 6x lim I 11 4x Use oo if the limit is and oo if the limit is

Suppose lim f x 8 Find the following Type DNE if the limit does not exist 48 2 lim 8 f x 8 2 lim 2 8 f x 8 2 lim z 8 f x 8

Suppose lim f x 7 lim g x 3 x x x f x 5x 3 3x 1 lim 4 xxxg x

3 If fcx 3x x and g x 3x 2 find and simplify each of time fonowing a g x 5 b fC3D S c gcf 2 D e

What radical function is represented in the graph f x Simplify your answer 9 3 0 4 4 1 5 X

Find the value s of k such that f x is continuous everywhere x 7 4x3 3kx 2 O f x x 2 x 2

13 Let f be the function defined by f x value of a is f continuous at x 10 x 14x 40 x 10 a x 10 For what x 10

Sketch the lines through the point with the indicated slopes on the same set of coordinate axes Point 3 5 S No Solution a 5 10 9 8 7 6 b 3 fa Slopes c d Undefined 10 2 2 3 4 5 6 7 00 40 W 5 89 10 Graph Layers Line 1 Point 1

Suppose f x has the following values X 5 9 f x 5 5 1 2 6 5 f x lim 100 81 000 3 810 Undefined 1 lim z 6 5 f x 1 3 8 0 8 1 lim z 6 5 f x appears to be Undefined 1 3 5 99 5 05 appears to be 5 999 5 005 appears to be 6 8 6 001 4 999 6 01 4 99 6 1 4 9

Evaluate the limit 5x 10x 7x lim 818 4 3x 6x Use oo if the limit is oo and oo if the limit is o

24 Given lim f x 6 and lim g x 6 evaluate x 1 x 1 lim 6f x 4g x x 1 lim 6 4f x x 1 lim x 1 Ag 2 6f x 4g x 6 4f x lim 6f x 4g x 6 4 x x 1

The path r t t i 21 1 j describes motion on the parabola y 2x 1 Find the particle s velocity and acceleration vectors at t 3 and sketch them as vectors on the curve The velocity vector at t 3 is v 3 1 12 J Simplify your answer including any radicals Use integers or fractions for any numbers in the expression The acceleration vector at t 3 is a 3 0 i 4 j Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Choose the correct sketch for the particle s velocity and acceleration at t 3 O A OB 2 Q O C

Suppose lim f x 17 and f x is nonconstant x 5 Provided the limit exists 7 12 1 f x 10 lim x 5 f x 17 Question Help Video A Written Example

Given lim f x 8 and lim g x 7 evaluate x 3 I 3 lim 6 f x 3g x x 3 lim 6 3f x I 3 lim 2 3 off 6f x 3g x 6 3f x 2 lim 6f x 3g x 6 3f x z 3

8 x x if x 4 if x 4 2x 3 Calculate the following limits Enter DNE if the limit does not exist lim f x Let f x lim f x 2 4 lim f x 2 4 Question Help Video B Written Example

Find the domain of the function Enter your answer using interval notation g x 4 7x 0

Suppose f x 7x 5 6 Evaluate lim f x x 5

Find the limit x 8x 15 lim 3 3x 91 10 Written Example

Evaluate the limit 4x 16 4 x 12x 32 lim Video 2 Writ

Suppose you know that lim f x 16 x 9 Find the following 16 f x lim x 9 f x 4 FA

Evaluate the limit lim y 0 1 8 y Y 1 64 11

Find the following values for f x sin I f x Based on these values lim sin I 0 7 0 0 0181 Olim sin 2 0 7 0 73354562801787 Olim sin 1 0 7 Be sure your computing device is set to RADIAN mod does not exist 0 00191 0 000103

Suppose lim f x 8 and lim g x 6 Find 24 7 24 7 lim 6 x 7 x 3g x 3g x 4x 4

Find the following values for f x sin X f x Be sure your computing device is set to RADIAN mode 0 1 0 01 0 001 Based on the values it appears lim f x I 0

Estimate the limit numerically or state that the limit does not exist 4 sin 9x lim 0 sin 8x Be sure your calculator is set in RADIAN mode and give your answer to at least three decimal pla Question Help Video Written Example

Find the following Simplify your answers completely f x 6x 7 g x 4 x a f g x b f g x c fg x d f g x What is the domain of f g Enter your answer using interval notation

Consider the functions below f x x 1 g x x 1 Find each of the following if possible If it is not possible enter NONE a fog b g f c fog 0

Evaluate the function for f x x 3 and g x x 2 f g 9 f g 9

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B whic 65 mi apart When the satellite is on one side of the two stations the angles of elevation at A and B are measured to 87 0 and 84 2 respectively Round your answers to the nearest mile 87 0 84 2 B a How far is the satellite from station 47 1324 mi 0 How high is the satellite above the ground

se the graph of h in the given figure to find the following values or state that they do t exist h 2 lim h x X 2 h 6 lim h x X 6 lim h x X 7 Type an integer or a decimal Find h 6 Select the correct choice and if necessary fill in the answer box to complete your choice A h 6 B h 6 is undefined Alim h x X 6 B d Find lim h x Select the correct choice and if necessary fill in the answer box to complete your choice X 6 Type an integer or a decimal Rich x OB The limit does not exist

mph of f to complete parts a through 1 If a limit ist explain why 5 answer box if necessary A 1 2 4 Type an integer or a fract OB The value of f 2 is undefir b Find lim f x Select the corre X 2 and fill in the answer box if necess A lim f x 3 X 2 Type an integer or a fractic B The limit does not exist bec defined for all x 2 c Find lim f x Select the correct X 2 and fill in the answer box if necessa OA lim f x X 2 Type an integer or a fractions B The limit does not exist becam defined for all x 2

PROBLEM 5 Find values 4p lim 2n 9n 3 9n 1 11448 3p lim 21 n 2 1 1 21 3p n 2 2 2 2 x 4 x 2 Csin