Limits & Continuity Questions and Answers

5 Verify that f and g are inverses f x x 7 g x 2x 14

uestion z 6 5 4 3 2 1 3 y 4 5 46 3 6 Write an equation for the function graphed above

Use a calculator to make a table to approximate the following limit Confirm your result with l H pital s Rule 17 1 X lim X 0 Complete the table below Do not round until the final answer Then round to six decimal places as needed 17 1 17 X 10 1 10 2 10 3 X h 10 1 10 2 10 3 X

Show all necessary work in your solutions Problem 1 Find the arc length of the graph of y x 2 3 2 from x 0 to x

Construct a table of values to find the value of the limit from both sides for lim x 2 x 8 2 x

Evaluate the following limits a b NOTE If needed enter INF for o and INF for o HINT Look at where the exponential fuction is going in the f 4 lim 200 ex 7 4 lim 2 0 e 7

The expression e7 9 6 can be written as ef a where f x is a function of x Find f x f x

A salesperson makes a base salary of 1100 per month Once he reaches 38 000 in total sales he earns an additional 6 commission on the amount of sales over 38 000 Write a piecewise defined function to model the salesperson s total monthly salary in as a function of the amount in sales x

What is the angle measure 52 48 51 equivalent to in decimal degrees Enter your answer rounded to the nearest thousandth of a degree in the box

Evaluate the function for f x x 3 and g x x 2 f g 1 f g 1

Find the following f x x 3 g x x 3 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 following graph 5 y 4 3 2 1 vertical shift of 3 units upward horizontal shift of 3 units to the left reflection in the x axis vertical stretch vertical shift of 3 units downward horizontal shift of 3 units to the right y Write an equation for the graphed function 5 4 3 Identify the parent function and describe the transformation shown in the graph Select all that apply 2 1 1 X

g x x 3 8 a Identify the parent function f f x b Describe the sequence of transformations from f to g A horizontal shift 3 units to the left a reflection in the y axis and a vertical shift 8 units downward A horizontal shift 3 units to the right a reflection in the x axis and a vertical shift 8 units upward O A horizontal shift 3 units to the left a reflection in the x axis and a vertical shift 8 units upward O A horizontal shift 3 units to the right a reflection in the x axis and a vertical shift 8 units downward O A horizontal shift 3 units to the left a reflection in the x axis and a vertical shift 8 units downward c Sketch the graph of g by hand H O 10 10 5 5 y 10 5 5 10 y 10 5 10 5 10 10 X X d Use function notation to write g in terms of the parent function f Og x f x 3 8 O g x f x 3 8 O g x f x 3 8 Og x f x 3 8 10 10 5 5 y 10 5 5 10 y 10 5 5 10 5 5

graph Determine which equation s may be represented by the graph shown Select all that apply Of x x 4 Of x 4 x Of x 4 Of x x 4 Of x x 4 Of x x 4

Compare the graph of the function with the graph of its parent function y x 4 Oy x 41 Oy x 4 O y x 41 O y x 41 Oy x 4 is f x x shifted up 4 units is f x x vertically stretched 4 fold is f x x shifted to the left 4 units is f x x horizontally stretched 4 fold is f x x shifted to the right 4 units

Determine the convergence or divergence of the sequence with the given nth term If the sequence converges find its limit Limit equal O Converges Limit equal 3 2 Converges Diverges None of the above an In n 2n

Let the function f be defined by the equation y f a where a and f x are rea numbers Find the domain of the function f x 49 4

oblem 6 Which of the following series converge Problem 6 M8 IM8 1M8 C n 6 6n 6 7n 1 gn iii elln n 1 A i and iii only B iii only C ii and iii only D all of them E i and ii only F 1 or G none of them H ii only

Foblem 5 Determine which of the following series converge 8T 5 2 4n n 1 n5 4n n8 8n 6 n 3n n n A none of them B iii only C i and iii only D all of them E i only F i and i G ii and iii only H ii only

0 1000 S 1546 1 in 1600 ft 1600 Bishop Peak 2002 Contour interval 40 ft FIGURE 17 18 If the distance between S and T on the map in Figure 17 is inch find each of the following a The horizontal distance between Stacey and Travis b The difference in elevation between Stacey and Travis c The angle of elevation from Travis to Stacey 19 Lompoc California is 18 miles due south of Nipomo Buellton California is due east of Lompoc and S 65 E from Nipomo How far is Lompoc from Buellton 20 A tree on one side of a river is due west of a rock on the other side of the

a b c d e f g h lim g t t 0 lim g t t 0 lim g t 2 lim g t t 2 lim g t t 2t lim g t t 2 g 2 lim 2 4 t

Use the given graph of f to state the value of each quantity if it exists If an answer does not exist enter DNE a b c d e f X 2 lim f x lim f x X 2 lim f x X 2 f 2 lim f x 4 X 4 2 f 4 0 4

The graph of f in the figure has vertical asymptotes at x 1 and x 5 Analyze the following limits a lim f x b lim f x X 1 X 1 e lim f x d lim f x X 5 X 5 Ay 10 c lim f x X 1 f lim f x X 5 Q Q G UA lim f x X 1 OB The limit does not exist and is neither oo nor 00 c Select the correct choice and if necessary fill in the answer box to complete your choice lim f x X 1 OB The limit does not exist and is neither oo nor co O A d Select the correct choice and if necessary fill in the answer box to complete your choice OA lim f x X 5 OB The limit does not exist and is neither o nor 00 e Select the correct choice and if necessary fill in the answer box to complete your choice OA lim f x x 5 OB The limit does not exist and is neither co nor co f Select the correct choice and if necessary fill in the answer box to complete your choice OA lim f x x 5 OR

3 Explain why sec is always greater than or equal to 1 or less than or equal to 1

nd exact values of the six trigonometric functions by hand for the following angle Do not use a T 4 e exact values of the six trigonometric functions for the given angle are below sin 4 T COS tan T CSC sec cot T 4 T 4 I II Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

1 cot 0 sin 0 1 Which of the following shows the key steps in establishing the identity OA 1 cot20 sin 0 sin 0 cot 0 sin 0 sin 0 cos 0 1 OB sin 0 O C O D 1 cot20 sin 0 csc 0 sin 0 sin 20 1 cot 0 sin 0 cos 0 sec cos 0 0 1 ec 0 1 cot20 sin 0 tan 0 sin 0 sin 20 S 1 cot 0 sin 0 1 cos 20 sin 20 cos 0 1 2 sin 0 in 0 sin 0 1

Find the slope of the secant line in red 510 to br p 09 ind 10 0 4 3 1 The slope of the secant line is 00 Undefined

Assume lim f x 10 lim g x 8 and lim h x 6 Compute the following limit and state the limit laws used to justify the computations X 9 X 9 X 9 f x lim x 99 x h x f x x 99 x h x Simplify your answer State the limit law s used to justify the computation Select all that apply lim A Difference B Quotient C Root D Sum E Power F Product G Constant Multiple D

1 cos 3x Hint What to do with that 1 cos 3x Maybe use conjugates Or maybe x if you rearrange one of the half angle formulas from Week 1 you get 2 sin 0 1 cos 20 could that help Actually either of these ideas will work so choose one d lim 20

In problems 29 32 use the information given in the diagrams to find angle A 43 44 45 A 4 20 6 8 30 A 4 46 A 3 10 2 45 A 2

Suppose that f x ln 5x 4 Find f x and use interval notation to give the domain of f Note When entering interval notation in WeBWork use I for o I for co and U for the union symbol If the set is empty enter without the quotation ma Domain

Evaluate the following two limits 1 lim x 2 2 lim x 2 1 x 2 1 x 2

2 Use the graph of each translated parent function to write its equation b O PERSEVERE Laura sketches the path of a model rocket that she launches a What type of function does the graph show 3 b In which axis has the parent function been reflected that on has Height m M 2814

Find the value of c where the table below probability distribution of random variable X X P X X 1 1 6 2 C Malm 4

Question 7 Suppose f x is continuous everywhere and has the following values 0 30 14 10 10 5 0 1 pt 3299 Details 20 10 I f x Based on this determine what you are willing to guarantee Between 10 and 2 20 you guarai Between 0 and x 10 you guarante Select an answer there is a root nothing Between 2 20 and z 30 you guarantee Select an answer Between z 30 and z 40 you guarantee Select an answer 40 20

The graphs of y f x solid and y g x dashed are shown 8 7 6 5 4 3 2 1 Find the following f 3 g 3 f g 3 fg 3 8 1 7 6 5 4 321 1 45 5 6 649 7 8 2 3 4 5 6 7 8

Suppose f is defined at x 2 and lim f a exists but lim f x f 2 x 2 x 2 The graph of y f z might look like select all that apply None of these Deta 20

Suppose f g are continuous functions where f 9 19 and lim 6f x 2g x 122 x 9 Then g 9 Question

Consider the graph shown Check all that apply lim f x exists x 2 a f 2 exists None of the above

2x 6 if x 4 Let f x 34 5x if x 4 if x 4 12 eta Determine whether f x is continuous at 4 If f x is not continuous identify why Not continuous f 4 is undefined Not continuous lim f x does not exist x 4 Not continuous lim f x f 4 x 4 The function is continuous at x 4

The graph of y f x has the following features lim f x 3 f x 2 f x f x xxx lim TH lim 24 1 lim 1 12 Which of the following might be a graph of y f z 8 78 8543 2 3 2 1 4 3 2 20 Z d a 9 9 4 3 2 3 16 112 84 4 5 6 3 2 8 7 6 4 2 F 2 12 4 4 9 b 8

suppose Iim f x x 2 The graph of y f x might look like select all that apply

Suppose f x IS The graph of y f x might look like select all that apply Q de x 1 CO

Consider the graph shown Check all that apply f 1 exists lim f x exists x 1 None of the above

Give a formula for a function f that satisfies lim f x o and lim f x 0 x 10 x 10 Choose the correct answer below X 1 X 10 OA f x OB f x x 10 x 11 OD f x x 10 x 1 OC f x 2 x 11 X 10

Choose the correct graph of the function y 2cos x Choose the correct graph below O A N Ay 10 G OB 10 Q O C Ay 10 Q OD 10 10

The graph below is the function f x 5 4 3 2 1 3 2 7 lim H 2 f 2 f x 2 123 4 5 lim f x x 2 lim f x x 2 1 1 Q Find the following If no value exists enter DNE C 2 3 4 5

lim sin x 4 sin 4 0 H 0 sin x 4 sin a cos 4 cos x sin 4 sin r sin 4 cos x cos 4 sin r sin 4 cos z cos 4 sin z cos 4 cos z sin 4 So sin 4 sin 4 So sin x cos 4 sin 4 cos r 1 sin r cos 4 cos x sin x sin x 4 sin 4 H 0 sin z cos 4 sin 4 cos z 1 sin 4 lim 2 0 H sin 4 H sin z 4 H 1 sin 4 H lim 2 0 sin x cos 4 K O lim 2 0 Part 2 of 5 Part 3 of 5 Part 4 of 5 Part 5 of 5 sin 4 cos z 1

lim sin 2 4 sin 4 2 0 sin x 4 sin z cos 4 cos z sin 4 sin z sin 4 cos z cos 4 sin a sin 4 cos z cos 4 sin z cos 4 cos z sin 4 So sin 4 sin 4 sin z cos 4 sin 4 cos z 1 sin z cos 4 cos z sin 2 sin z 4 sin 4 F So 0 sin z cos 4 sin 4 cos z 1 Um z in 4 z z lim sin z cos 4 lim Part 2 of 5 Part 3 of 5 Part 4 of 5 Part 5 of 5 sin 4 cos z 1

Suppose a student finds the following values of x and f x x If f X 1 9 10 1 99 344 1 999 8197 L these values would make us DOUBT the claim that lim f x x 2