Limits & Continuity Questions and Answers

Write a in the form a T aNN without finding T and N r t g cos t i g sin t j htk

8 Give the precise definition of saying a function f r is continuous at a number x a Explain how a function may fail to be continuous at x a Illustrate each case with a picture 9 How can a function f x fail to be differentiable at a number r a Explain all cases and

Find the perimeter of the triangle whose vertices are 1 2 1 6 and 4 1 Write the exact answer Do not round

f r t f t dt where f is the function whose graph is shown Let g x g 12 g 15 g 18 11 y 9 6 a Evaluate g x for x 0 3 6 9 12 15 and 18 g 0 g 3 g 6 g 9 minimum 3 3 3 f 6 9 12 15 18 21 t b Estimate g 21 Use the midpoint to get the most precise estimate g 21 c Where does g have a maximum value Where does it have a minimum value

Use L Hospital s rule to determine the following limit Use exact values e r 1 lim 2 0 0 sin 4x

Evaluate the definite integral Lt 5 1 t 5 t dt T 2 la

9 Find the limit if it exists lim f x x 1 9 b 0 c 4 d 5 e 12 where a Limit does not exist f x x 4 x 1 x 4 x 1

lim 8 Find Ax 0 f x Ax f x Ax a 0 b 3 c 5 d 1 e Limit does not exist where f x 3x 5

1 Let lim f x x 1 f x Determine the following limit a 4 b 3 c 2 d 0 3 x x 1 0 x 1 e does not exist

Determine whether the following function is continuous at x 0 Explain mathematically f x 1 if x 0 0 if x 0 f 0

Find the limit lim t 0 e 6ti 2 sin t j sin 7t k

Determine the vertex of the parabola whose graph 654321 Provide your answer below 6 5 4 f 7 6 5 4 3 2 11 1 2 3 4 5 6 7 1 2 1 2 ITT 123 3 4 st 106 5 456 6 Z Give your answer as an ordered pair h k For example if you found the vertex was at 1 2 you would enter 1

Use a table of values to estimate the value of the limit correct to three decimal places If you have a graphing device use it to confirm your result graphically sin 50 lim 0 0 tan 20

The difficulty in finding the velocity after 3 seconds is that we are dealing with a single instant of time t 3 so no time interval is involved However we can approximate the desired quantity by computing the average velocity over the brief time interval of a tenth of a second from t 3 to t 3 1 average velocity Time interval 3 t 4 3 t 3 1 The following table shows the results of similar calculations of the average velocity over successively smaller time periods 3 st 3 05 3 t 3 01 change in position time elapsed 3 t 3 001 s 3 1 s 3 0 1 9 4 9 0 1 Average velocity m s 34 3 29 89 29 645 m s 29 449 29 4049 m s rounded to one decimal place The instantaneous velocity when t 3 is defined to be It appears that as we shorten the time period the average velocity is becoming closer to limiting value of these average velocities over shorter and shorter time periods that start at t 3 Thus the instantaneous velocity after 3 s is the following Round your answer to one decimal

Determine the infinite limit 8 x 7 6 lim X 6X

The function f x Using a calculator and remembering that if x ER sin x means the sine of the angle whose radian measure is x we construct a table of values correct to eight decimal places X 1 0 sin x is not defined when x X 0 3 0 5 0 95885108 sin x X 0 4 0 97354586 0 1 0 84147098 0 2 0 99334665 0 98506736 0 001 0 99833417 10 05 0 99958339 0 01 0 99998333 0 005 0 99999583 0 99999983 From the table and the following graph we guess that lim X 0 X y sin x sin x

A graphing calculator is recommended X x 0 1 3x 1 a Estimate the value of lim by graphing the function f x X 1 3x 1 Round your answer to one decimal place b Make a table of values of f x for x close to 0 and guess the value of the limit Round your answer to six decimal places c Use the Limit Laws to prove that your guess is correct Find the exact value

We do so by evaluating the function f h 2 h 4 16 for various values of h close to the left side and the right side of h 0 Intuitively we do this because then we should be able to h find a value of f h as close to the limit as we like if we choose h sufficiently close to 0 Complete the tables for f h evaluated at the given values correct to six decimal places h 0 5 0 1 0 01 0 001 lim 2 h 4 16 h 0 h 0 0001 f h 34 481000 32 240801 h 0 5 0 1 0 01 0 001 0 0001 f h 21 875000 31 760799

Use the graph of the function f to state the value of each limit if it exists If an answer does not exist enter DNE f x x 9 x 2 a lim f x x10 b lim f x x 0 c lim f x X 0

Is there a number a such that the following limit exists If an answer does not exist enter DNE 2x ax a 6 x x 2 lim X 2 Find the value a a 11 Evaluate the limit

Determine the infinite limit O 8 O 88 lim In sin x x 0

Evaluate the expression without using a calculator 34 If you have had difficulty with this problem you may wish to consult the Review of Algebra on the website Stewar

re 1 Use the data table below and the attached graph paper to plot the mass and volume of the 5 samples of the minerals galena below Note the resulting line you plot is the minerals density 2 Calculate the density of samples 1 5 and place the value in the density column of the data table below 3 Answer the questions below Sample 1 2 3 4 5 Size small large 9 Mass 15 g 60 g 120 g 480 g 750 g X Volume 2 cm 8 cm 16 cm 64 cm 100 cm Density d m v 7 5cm 7 554 T ser n Questions 1 Use the completed graph to determine how much mass a sample of galena would have if its volume was 75 cm 22 2 Use the completed graph to determine how much volume a sample of galena would have if it s mass was 300 g 4 How does the density of Sample 2 compare to that of Sample 4 3 Describe the relationship between mass and volume shown by the graph 6 What is the effect of sample size on the density of a material 5 How did the density of the largest sample 5 compare to the smallest sample 1 7 Describe the trick you were taught on how to change around the density formula to solve for different parts of it

1 Find the following 11 6 a sin b tan 300 c cos570

Find the equation of the line through the points 1 7 and 2 1 Enter your answer in slope intercept form Provide your answer below

Determine whether the series converges absolutely or conditionally or diverges 8 1 n n 1 O converges conditionally O converges absolutely O diverges

a Graph the given function b Find all values of x where the function is discontinuous if x 3 g x x 1 if 3 x 1 if x 1 2 2 a Choose the correct graph below O A 10 Q G OB 15 Q b Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function is discontinuous at x Use a comma to separate answers as needed OB The function is continuous for all values of x O C 15

Is the function continuous at the given values of x 1 x x 9 h x x 0 x 2 x 9 Is the function continuous at x 0 O No Is the function continuous at x 2 O No Is the function continuous at x 9 O No

Use a calculator to estimate the limit 1 lim 2 x 0x 3x OA 00 OB 0 O C Doos pot

Suppose lim f x 6 and lim g x 2 Use the limit properties to find the given limit X 5 X 5 lim X 5 f x g x 6g x Apply limit properties to the given limit Choose the correct answer below f x g x X 5 6g x OA lim f x g x OB lim X 5 6g x OC lim X 5 lim X 5 OD lim X 5 f x g x 6g x Find the limit f x g x 6g x 11 0 lim f x lim g x X 5 g x X 5 f x g x x 5 6g x lim f x lim g x X 5 X 5 lim 6 lim g x X 5 X 5 lim f x lim g x x 5 lim 6 lim g x X 5 X 5 lim f x lim g x X 5 lim 6 km 1 X 5 X 5 lim g x X 5 9 x xim lim 6 lim g X 5 X 5 g x

Determine the values of a and b which would result in the function f x being differentiable at x 5 f x 3ax 2 2bx 6x 4 for for x 5 x 5

a Using a coefficient of 2 for x the equation for the tangent plane is b Find the equations for the normal line Let x 2 4t 0 y 0 z 0 Type expressions using t as the variable

Given that Cos 0 3 QII for find the Stry Funchan

The container of a breakfast cereal usually lists the number of calories and the amounts of protein carbohydrate and fat contained in one serving of the cereal The amounts for two common cereals are given below Suppose a mixture of these two cereals is to be prepared that contains exactly 295 calories 9 g of protein 48 g of carbohydrate and 8 g of fat a Set up a vector equation for this problem Include a state ment of what the variables in your equation represent b Write an equivalent matrix equation and then determine if the desired mixture of the two cereals can be prepared Nutrition Information per Serving General Mills Cheerios 110 4 20 2 Nutrient Calories Protein g Carbohydrate g Fat g Quaker 100 Natural Cereal 130 3 18 5

A graphing calculator is recommended For the limit lim x35x 9 21 illustrate this definition by finding the largest possible values of that correspond to 0 2 and 0 1 Round your answers to four decimal places X 3 0 2 0 1 8 8 X X

Question Each answer choice below represents a relation by a set of ordered pairs In which of the answer choices is the relation function Select all correct answers Select all that apply 2 1 1 3 5 9 8 1 2 1 2 1 2 12 4 7 9 2 4 13 0 7 3 13 7 2 2 9 1 6 1 6 5 10 6 9 7 2 6 5

Use the graph 19 3 m 76 17 f x x C 9 LE 1 T L U LE 11 1 LE LE E E LE 10 L 3 1 6 289 9 36 y x 361 36 Ly

5 6 Let z x iy and f z x y 0 2xy i x y x y Find lim f z if it exists or show that the limit doesn t exist 2 0 if z 0 if z 0 Let f be the function defined in problem 5 Find f z if it exists or show that it doesn t ex

Evaluate the following limit lim x 12 1 99 2 1 2 I 0 A

Evaluate the following limit lim 25 X 5 2 5 10 Does not exist undefined 0

Evaluate the following limit x 25 lim 2 5 5 0 0 AT Does not exist undefined

1 To carry a suitcase on an airplane the length width height of the box must be less than or equal to 62 inches Assuming the base of the suitcase is a square show that V n 31 2 h What height allows you to have the largest volume

Is the following statement x 2xy Let f x y x 4y 0 true or false if x y 2 1 if x y 2 1 Then f x y is not continuous at the point 2 1

How many bit strings of length 7 contain at least three 1s O 100 35 99

What is the remainder when 3x 3x 8x 37x 77 is divided by x 3 Only enter a number

Sketch the graph of a function that satisfies all these conditions Sketch your graph on paper and upload a PDF or image of your solution lim f x 0 lim f x 1 f 1 is undefined lim f x 3 lim f x 1 f 3 lim f x does not exist f 1 1 x 1 4 x 14 x 3 3 x111

If it exists find lim f x x 3 f x g x provided that lim f x 1 lim f x 3 f 2 6 lim g x 7 lim g x 5 g 2 2 x 2 1 2 x 2 Write your solution on paper and upload a PDF or image of your solution Explain how you solved the problem and why you solved it like this

5 points Suppose f x x x 2 Use the definition of the derivative to find f 1 What does f 1 represent here Sketch the graph of this function draw the tangent line to the graph a the point 1 0 and show four secant lines that are approaching this tangent line Write your complete solution on paper and upload a pdf or image of your solution Attach File Save

13 Determin whether y is function of x y 3x 1 a no b ye S

11 Let a 15 b 14 c 19 d 11 f x 14x 10 undefined Then simplify the expression f x 10 x 10