Limits & Continuity Questions and Answers

The following limit is a definition of f a Determine the function f x and 6 h 6 66 h lim h 0 The function is f x

Find the limits below lim 2sin x lim x0 et 1 lim 10 1 lim 1 0 5 5 cos x et x 1 t In 1 2t 1 im t

2 10 pts Find the slope of the tangent line to y 2 x at x 0 Hint we must use a limit here to get the exact value and then some algebra remember the conjugate

2 Draw a graph of a function that has the following characteristics f 2 0 f x 0 for x 2 f x 0 for 2 x 1 limx 1 1 limx 1 3 f x 0 for all x 1 lim C4

The graph to the right shows the closing share prices in dollars for Company A and Company B on the 21 trading days of the month of January 2018 What was the approximate closing share price for Company A on day 21 Company B The closing share price for Company A on day 21 was 15 5 Round to the nearest ten cents as needed The closing share price for Company B on day 21 was Round to the nearest ten cents as needed 25 24 23 22 A Price 21 M 20 19 18 17 16 15 14 M Company A 25 24 23 Company B 22 21 13 0 2 4 6 8 10 12 14 16 18 20 22 20 19 18 17 16 15 14 400 Day C G E

2 Using the graph of f x and the definition of continuity a Is the function continuous at x 0 Why or why not b Is the function continuous at x 2 Why or why not c Is the function continuous at x 6 Why or why not f x N 20 3 Consider the function f x X x 6 a Does the Intermediate Value Theorem guarantee a root zero of the function on the interval 0 8 Why or why not b Does the Intermediate Value Theorem guarantee a root zero of the function on the interval 4 0 Why or why not

Find the x intercept s and y intercept of the graph of the equation y 10x 6x 5 What are the x and y intercepts OA There are zero x intercepts and a y intercept at OB There is one x intercept at and a y intercept at OC There are two x intercepts at and and a y intercept at

z x Examine z dy for extremum the function 3 r 6ru

curve is given by y L x where 0 x 1 represents the lowest fraction of the population of a society in terms of wealth and 0 sys1 represents the fraction of the total wealth that is owned by that fraction of the society Use this information to complete parts a through g Click the icon to view an example of a Lorenz curve 0 2A 1 Therefore G 2A 1 2 L x dx by the 0 Simplify transitive property completing the proof e Compute the Gini index for the cases L x xP and p 1 3 1 9 2 3 4 The Gini index for p 1 3 is 0 130 Type an integer or decimal rounded to three decimal places as needed The Gini index for p 1 9 is 0 310 Type an integer or decimal rounded to three decimal places as needed The Gini index for p 2 is 0 333 Type an integer or decimal rounded to three decimal places as needed The Gini index for p 3 is 0 500 Type an integer or decimal rounded to three decimal places as needed The Gini index for p 4 is 0 600 Type an integer or decimal rounded to three decimal places as needed f What is the smallest interval a b on which values of the Gini index lie for L x xP with p 1 Points 0 52 of 1 points Save

The speed of a runner increased steadily during the first three seconds of a race Her speed at half second intervals is given in the table In this problem we are going to estimate the distance that she traveled during these three seconds 3 t seconds 0 0 5 1 1 5 2 2 5 v ft s 0 6 3 11 3 14 6 18 2 19 4 20 1 Speed of the Runner Left Endpoint Approximation Estimate the distance traveled by the runner using left endpoints Distance Right Endpoint Approximation Estimate the distance traveled by the runner using right endpoints Distance feet feet 4

nd all values of x such that f x 0 Enter your answers as a comma separated f x 4x 7

Evaluate the function at each specified value of the independent variable and simplify If an answer is undefined enter UNDEFINED f x 9 a b 9 2x c 8x 1 f 4 f 0 f 1 x 0 0 x 1 x 1

valuate the function at each specified value of the independent variable and simplify If an answer is undefined enter NDEFINED q t a b c 6t 7 9 2 q 0 q x

Refer to the figure below to find the limit lim f x X 0 4 5 0 5 0 5 Q Select the correct choice below and fill in any answer boxes in your choice OA lim f x x 0 OB The limit does not exist

Refer to the graph at the right to find the limit lim 4 f x X 8 Select the correct choice below and fill in any answer boxes in your choice OA lim 4 f x 811 Simplify your answer B The limit does not exist 4 2

Use limits to compute f x f x f x 9x X 2

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 d lim X 4 f x x 4f x g x a Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x 3g x X 4 Type an integer or a simplified fraction OB The limit lim f x 3g x does not exist X 4

Question 1 14 points Consider the function f x 2x 6 2x r3 a Factor the numerator and the denominator of f x as much as you can b Write the domain of f c Say whether f has any symmetries d Find the intersections of the graph of f with the z axis and the y axis if any e Find the subset of the domain on which f is positive f Compute the limits of f at any interesting point in R and at too Write the equation of any vertical horizontal or oblique asymptote of the graph of f g Compute the derivative f x h Find the stationary points of f if any Find the subsets on which f is increasing and decreasing Say whether the stationary points are relative minima maxima or neither Compute the values of f x at the stationary points Recall to factor a polynomial of the form ax bx c substitute t x so that it becomes at2 bt c and factor this first Then substitute back r for t and factor again i Draw the approximate graph of f encoding all the information obtained above

For the following exercises consider the function f x x 1 x 1 30 T Complete the following table for the function Round your solutions to four decimal places X 0 9 0 99 0 999 0 9999 f x response a b C P d X 1 1 1 01 1 001 1 0001 f x e f g h 31 What do your results in the preceding exercise indicate about the two sided limit limf x Explain your x 1

the following exercises set up a table of values to find the indicated limit Round to eight digits x 4 x 2x x 6 38 lim X 1 9 1 99 1 999 1 9999 x 4 x x 6 a b C d X 2 1 2 01 2 001 2 0001 x 4 x x 6 e f g h

Find the area of the following region expressing your result in terms of the positive integer n 2 The region bounded by f x x and g x x n for x 0 The area of the region in terms of n is

Use a calculator to approximate sin 5 8916 and cos 5 8916 sin 5 8916 Round to four decimal places

estion What is the net signed area between f x 2x 3 and the x axis over the interval 4 2 Do not include any units in your answer Provide your answer below 11262

Question Find the total area between the function f x 2x and the x axis over the interval 3 3 Provide your answer below

Calculate the following limits at infinity If necessary use inf and inf to denote o and o Use DNE if a limit does not exist 1 lim sin x dne 818 2 lim sin x dne 8118 3 lim tan x 818 4 lim tan x 118 TC 2 TC 2 Part 2 Horizontal Asymptotes List any horizontal asymptotes for the following functions If there are no asymptotes enter NONE If there is more than one asymptote give a comma separated list i e 1 2 1 The function sin x has horizontal asymptote s y none 2 The function tan x has horizontal asymptote s y TC 2

Calculate the following limits at infinity for the given function f x If necessary use inf and inf to denote and Use DNE if a limit does not exist 1 lim f x 0 2 lim f x 818 1 200 f x 1 e has horizontal asymptote s y 0 1 Part 2 Horizontal Asymptotes List any horizontal asymptotes for this function f x If there are no asymptotes enter NONE If there is more than one asymptote give a comma separated list i e 1 2 1 The function f x 2

Calculate the following limits at infinity for the given function f x If necessary use inf and inf to denote and Use DNE if a limit does not exist 1 lim f x f x 4x x 15 x1 5 3x Part 2 Horizontal Asymptotes List any horizontal asymptotes for this function f x If there are no asymptotes enter NONE If there is more than one asymptote give a comma separated list i e 1 2 4x x 15

Find the circular function values for the given number s State which functions are undefined for s Do not use calculator 17x 2 SE COLX Select the correct choice below and if necessary fill in the answer box to complete your choice 1 0 Ay 2 0 2 0 1 0

Determine the point on the graph of the unit circle that corresponds to 2x Then findcos 2x andsin 2x and state which functions are undefined for 2 The point on the unit circle is Typo an ordered pair Type an integer or a simplified fraction G 1 0 R 2 Ay 0 1 0 0 1 1 0 A o

Consider trying to evaluate the integral A A E B A ln x 1 O C Aarctan r A D x 4x 3x x 2 x 2 OE A ln x If you asked a computer algebra system for the answer which of the following terms would you know is wrong if it showed up in the answer it gave you WITHOUT even solving for the coefficients dx

onvert the following degree measure to radians 120 120 radians Simplify your answer Type an exact answer using a

14 lim x 3 2 x 1 X 1

4 lim x 2 x2 5x 6 x 2

11 lim x 0 x 3 3

x 1 7 lim x 1x 1

8 lim X 2 x 3x 2 x 4

2 lim x3 2x 3x 3 x 1

1 lim 3x 2x 1 x 2

The line y 5x b is tangent to the graph of y x at the point P a a Find P and determine b P b 1 100 10 Type an ordered pair Type an Integer or simplified fraction

Apply the three step method to compute the derivative of f x 2x f x

Find the point s on the graph to the right where the slope is equal to 6 11 The point s on the given graph at which the slope is equal to 11 is are 2 2 11 11 1 y x3 x y X Slope is 3x E 2 11 ad Uco a comma to separate answers as needed

Find the point on the graph of y x where the curve has slope m 5 The point on y x at which the tangent line has slope 5 is Type an ordered pair Use integers or fractions for any numbers in the expression

Convert the following radian measure to degree measure 4x 4x 3 11

Find the complementary and supplementary angles to 12 T 12 12 Simplify your answer Type a fraction with in the numerator The complementary angle to 5x The supplementary angle to is 5x 12 Simplify your answer Type a fraction with in the numerator is

Directions Consider the following function f x at a 3 Part 1 Difference Quotient A If h0 then the difference quotient can be simplified into the form f 3 h f 3 h where A B C D and E are all non negative constants Find these constants B C 0 0 D Note It s possible for one or more of these constants to be 0 E 5 x A Bh C Dh Eh A Bh C Dh Eh f 3 lim h 0 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 3 That is note the negative in front f 3 h f 3 h

Use the graph of f x given below to evaluate the following limits at infinity 1 lim f x 2 x 2 lim f x If necessary use inf and inf to denote o and co and DNE if the limit does not exist

region in two ways a Using integration with respect to x b Using geometry OC 2 6 x dx 0 OD 6 S 0 6 x x dx b Using geometry which would be the appropriate set up for the area OA A 2 6 3 1 OB A 1 2 6 6 6 3 c A OD A 2 6 6 The area of the shaded region is

Tutorial Exercise Use the graph to determine the x values at which fis discontinuous For each x value determine whether f is continuous from the right from the left or neither 2 2 Submit Skip you cannot come back 4 6 Step 1 To find the numbers for which is discontinuous we look for x values for which the function is not defined or the left and right limits do not match Starting from the left and moving right the first x value for which f x is discontinuous is x At this point f x is not defined Therefore at this point f is which of the following O continuous from the right O continuous from the left O neither

3 Six graphs of functions are below along with six graphs of derivatives Matell of each function with the graph of its derivative Original Functions X 4 A D 34 Their derivatives B E X W F

Find the x value at which fis discontinuous and determine whether fis continuous from the right or from the left or neither ifx so 6 x 8 x x 8 if 0 x 8 if x 8 a Find the x value at which is discontinuous X f x 8 x b Determine whether f is continuous from the right or from the left or neither O continuous from the right O continuous from the left O neither c Sketch the graph of f Select the correct graph f x 10 10 5 10 5 10 f x f x 10 AAN 10 5 10 10 10 10 5 10 5 00 00 10 5 0 5 10