Limits & Continuity Questions and Answers

etch a possible graph of a function that satisfies the conditions below f 0 3 lim f x 0 lim f x 3 X 0 X 0 hoose the correct graph below A B Q O C Q O D

jecture f x 1 cos 5x 5 5 x 1 x a a 1 e a graphing utility to graph f Select the correct graph below Q A n ch graph is displayed in a 1 3 by 0 5 window B Wh C EXIST O D Ins m

ketch a graph of f and use it to make a conjecture about the values of f a lim f x lim f x and lim f x or x a x a x a they do not exist f x 1 X 1 A ketch a graph of f Choose the correct graph below 10 a 1 T Q Q S B 10 Q O C 10 A The value of f 1 2 Type an integer or a decimal OB The value of f 1 does not exist Q Q Q D 10 lake a conjecture about the value of f 1 Select the correct choice below and if necessary fill in the answer b Omplete your choice

the graph of f in the figure to find the following values or state that they do exist If a limit does not exist explain why 3 im f x im f x m f x 3 3 ind the function value Select the correct choice and if necessary fill in the answer box within your A f 3 1 Type an integer or a fraction B The value of f 3 is undefined ind the limit Select the correct choice and if necessary fill in the answer box within your choice Alim f x 1 Type an integer or a fraction X 3 B The limit does not exist because f is not defined for all x near 3 with x 3 ind the limit Select the correct choice and if necessary fill in the answer box within your choice A lim f x Type an integer or a fraction X 3 B The limit does not exist because f is not defined for all x near 3 with x 3

A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two dimensional ice cream cone as shown in the figure P 10 cm A 0 B 0 01 R Q 10 cm If A 0 is the area of the semicircle and B 0 is the area of the triangle find A 0 lim 0 0 B 0

3 2 Sketch a function that meets all of these conditions a b lim f x 2 814 C lim f x 4 X1 8 e f lim f x co 8 X 3 d lim f x 2 X 3 f 3 5 lim f x 7 X 0 g f 0 1 9 8

On June 1 20Y2 Herbal Co received 25 830 for the rent of land for 12 months Journalize the adjusting entry required for unearned rent on December 31 20Y2 Round your answers to the nearest dollar amount If an am does not require an entry leave it blank Dec 31 Unearned Rent

In the figure the straight line is tangent to the graph of f x Find f 2 and f 2 f 2 Type an integer or a fraction Simplify your answer 8 0 2 2 y f x

At the end of the holiday season in January the sales at a department store are expected to fall It is estimated that for the x day of January the sales will be 16 S x 3 thousand dollars x 1 a Find the total sales for January 9 and determine the rate at which sales are falling on that day b Compare the rate of change of sales on January 3 to the rate on January 9 What can you infer about the rate of change of sales a The total sales for January 9 were thousand dollars Sales y 4 Days in January

Find the functions and their domains Enter the domains in interval notation x 5 x 2 f x x a fog fog x domain b gof X gof x domain fof fof x domain d gog gog x domain g x

How many significant figures are there in each of the following numbers 63 43 0 08852 177 24 0 823958

The boiling point of a liquid is calculated to be 311 K What is this temperature on the Celsius and Fahrenheit scales F

Use a calculator to find an approximation to each of the following expressions Round to four decimal places 51 sin 60 53 V3 55 sin 45 57 1 2 52 cos 30 54 V 56 cos 45 58 3

Evaluate the following limit Use l H pital s Rule when it is convenient and applicable y 5y 36 y 4 32 y y lim lim y 5y 36 y 4 32 y y 11 Type an exact answer

2 4 pts 4 pts Sketch the graph of the function f x the asymptotes of the graph using dashed lines x 4 2 3 Clearly show X 4

Find the circular function values for the given number s State which functions are undefined for s Do not use calculator S 17x 2 OA cos OB The function is undefined Select the correct choice below and if necessary fill in the answer box to complete your choice 17x 2 Type an integer or a fraction 1 0 2 The function is undefined Ay 2 0 0 1 P 1 0 0 1 2x Q S A

Determine the values of the trigonometric functions of a where sin a 25 can a CSC sec a 7 24 25 7 Type a simplified fraction Type a simplified fraction Type a simplified fraction and cc

The function graphed is of the form y a sin bx or y a cos bx where b 0 Determine the equation of the graph y Type an expression using x as the variable 0

The function graphed to the right is of the form y a sec bx c or y a csc bx c for some a 0 b 0 Determine the equation of the function C 10 Ay 8 6 4 112 77 LS Lat 817 10

Find the a period b phase shift if any and c range of the following function 1 X 3 5 y y 3 cot I 5 a The period is Type an exact answer in terms of

The function graphed to the right is of the form y a tan bx or y a cot bx for some a 0 b 0 Determine the equation of the function Half and quarter points are identified by dots An equation of the function shown is y 10 8 6 4 6 Ay 10

41 sin 0 cc

Multiply 21 sin 4 sin 0 3 23 2 cos 0 3 4 cos 0 5 25 1 sin 1 sin 0 22 cos 2 cos 0 5 24 3 sin 0 2 5 sin 0 4 26 1 cos 0 1 cos 0

40 Suppose f is an odd function and is differentiable everywhere Show that fo every positive number 6 there exists a number cin b b such that f b f c b

Determine if the function f x is continuous and or differentiable at x 3 f x x 18 for 3 x 3 7x for 3 x 8 Is f x continuous at x 3 O Yes No

Determine the value of a that makes the function f x continuous at x 9 x 45 x 9 x29 f x ax Select the correct choice below and if necessary fill in the answer box to complete your choice O A a

Determine whether the following function is continuous and or differentiable at x 10 f x x Is the function continuous at x 10 OA Yes the function is continuous at x 10 OB No because while the function is defined at x 10 and lim f x exists lim f x is not equal to the value of the function at x X 10 X 10 OC No because the function is not defined at x 10 OD No because while the function is defined at x 10 lim f x does not exist X 10

Determine whether the following function is continuous and or differentiable at x f x Is the function continuous at x 6 OYes 2x 6 for 0 x 6 6 for 6 x No

Is the function graphed to the right continuous at the point x 1 Choose the correct answer below 54 3 2 1

is the average value of f x on 0 x 2 alue is the average value of g x on 0 x 2 alue is the average value of f x g x on 0 x 2 alue e following statement true 0 5 0 5 0 5 0 5 f x 015 2 0 5 0 5 F 8 5 1 1 g x 1 1

x 1 2x 1 7 f x 2 1 x 4 1 2 2 x 3x a at x 1 x 4

8 g x 2x 1 In x a We x 3 3 x 1 x 1 x 3x 7 at x 3 raping ant sau b at x

3 f x X x 2x

Sketch the graph of the piecewise defined function x 7 3x 5 f x 20 20 x 4 X 4 y 20 10 1 0 20 y 20 10 20 10 10 20 20 X X 20 20 y 20 10 20 y 20 10 20 10 10 20 20

Sketch the graph of a function with the given properties You do not need to find a formula for the function h 2 1 lim h x 4 lim h x 0 h 1 lim h x 3 lim h x 2 X 2 X 2 X 1 X 1 Which of the following is a graph that satisfies all of the given conditions A O C A Q 5 o S OB O D Q Q G Q

The median player salary for the New York Yankees was 1 5 million in 2007 and 1 7 million in 2013 Write a linear equati giving the median salary y in terms of the year x Let y be in millions of dollars and let x 7 represent 2007 Use the equation to predict the median salary in 2027 Round your answer to one decimal place million

Evaluate lim 7x 3x 5x X 0 X If an answer does not exist enter DNE 8 CO X

49 x x 7 x 7x If an answer does not exist enter DNE Evaluate lim

Limits Continuity Course Packet on computing limits from graphs The graph of y f x is shown below 1 y 2 1 2 2 3 Use the graph of y f x to compute each of the following values If an answer does not exist e f 1 DNET Enter a number lim f x DNE

Find the volume of the solid generated when R shaded region is revolved about the given line y 1 x y 1 x 1 about y 1

The solutions x y of the equation x2 16y2 16 form an ellipse as pictured below Consider the point P as pictured with x coordinate 2 a The ellipse is not given by a single function as it fails the vertical line test It is given by two function The points P and Q shown are in the upper part which can be described as y f x were f x b Let h be a small non zero number and form the point Q with x coordinate 2 h as pictured Then the y coordinate of Q is given by c Let h be a small non zero number and form the point Qwith x coordinate 2 h as pictured The slope of the secant line through PQ denoted s h is given by the formula d Rationalize the numerator of your formula in c and simplify to rewrite the expression so that it looks like f h g h subject to these two conditions 1 the numerator f h defines a line of slope 1 2 the function f h g h is defined for h 0 When you bu finished your simplification neither the numerator nor the denominator should become zero when h 0 When you do this 1 h g h e The slope of the tangent line to the ellipse at the point P is lims h 110

This worksheet reviews limits and the definition of the derivative with graphs and computations 1 Answer the following questions using the graph y f x below The function f x has domain all numbers except 7 as seen from the graph a lim f x b lim f x c f 0 d lim f x e lim f x x 0 x f lim h 0 15 14 13 12 11 101 761 g f 5 f 3 h 5 h 25 20 15 10 3 10 h lim h 0 j lim h 0 10 11 12 13 14 k lim h 0 f 8 h f 8 h f 8 h h f 6 h f 6 h f 3 h 5 h 1 List all the intervals where the derivative f x is negative m List all the intervals where the derivative f x is decreasing n A critical value for f x is any z in the domain of f x where f x 0 or f x is undefined List all critical values of f x

Worksheet Math 124 2 Evaluate the following limits and then match the functions with their graphs shown below using your limit results Some will require you to compute left and right hand limits 1 a lim 1 57 5 b lim T H x 5 x 2x 35 x 5 x 4x 5 c lim d lim x 5 x 3x 10 x 5 Week 4

B 15 Let m and b be real numbers and let the function fbe defined by f x 1 3bx 2x for x 1 mx b for x 1 Iffis both continuous and differentiable at x 1 then A m 1 b 1 B m 1 b 1 D m 1 b 1 2 C m 1 b 1 E none of these

Guess the value of the limit lim x18 by evaluating the function f x for x 0 1 2 3 4 5 6 7 8 9 10 20 50 and 100 Use a graph off to support y

Find the horizontal and vertical asymptotes of the curve You may want to use a graphing calculator or computer to check you DNE X 11 y 4x x 1 x x 6

Find the horizontal asymptote y 3x 6 X E f x

Suppose that lim f x 9 and lim g x 3 Compute the limits that exist X 4 X 4 a lim f x 3g x a Select the correct choice below and if necessary fill in the answer box to complete your choice A lim f x 3g x 0 X 4 Type an integer or a simplified fraction OB The limit lim f x 3g x does not exist X 4 b lim f x g x c lim 4f x g x d lim X 4 X 4 X 4 b Select the correct choice below and if necessary fill in the answer box to complete your choice A lim f x g x 27 X 4 Type an integer or a simplified fraction OB The limit lim f x g x does not exist X 4 A c Select the correct choice below and if necessary fill in the answer box to complete your choice OB The limit lim 4f x g x does not exist X 4 OA lim 4f x g x 108 X 4 Type an integer or a simplified fraction OB d Select the correct choice below and if necessary fill in the answer box to complete your choice f x f x g x f x g x lim X 4L Type an integer or a simplified fraction f x f x g x X 41 The limit lim does not exist

Determine if the limit below exists If it does exist compute the limit x 3x 18 10 2x lim X 10 Select the correct choice below and if necessary fill in the answer box to complete your choic OA C B 3x 18 lim x 10 10 2x Type an exact answer using radicals as needed The limit does not exist

Suppose that lim f x 9 and lim g x 3 Compute the limits that exist X 4 X 4 a lim f x 3g x X 4 b lim f x g x X 4 c lim 4f x g x X 4 OB The limit lim f x 3g x does not exist X 4 a Select the correct choice below and if necessary fill in the answer box to complete your choice A lim f x 3g x 0 X 4 Type an integer or a simplified fraction OB The limit lim f x g x does not exist X 4 f x x 4f x g x d lim x b Select the correct choice below and if necessary fill in the answer box to complete your choice A lim f x g x 27 X 4 Type an integer or a simplified fraction OA lim 4f x g x X 4 Type an integer or a simplified fraction OB The limit lim 4f x g x does not exist X 4 c S lect the correct choice below and if necessary fill in the answer box to complete your choice