Limits & Continuity Questions and Answers

2 Use the graphs of f x and g x to construct the graph of h x f g x on the grid provided 4 TA K g x 3 Math Kangaroo is an international mathematics competition that is held in over 400 countries including Canada Suppose the cost of preparing booklets for the Canadian version of the contest includes 675 in fixed costs and 3 50 per booklet The booklets are sold for 30 each Write simplified equations that represent a The total cost C as a function of n the number of booklets produced b The revenue R as a function of n the number of booklets produced c The profit P which is the difference between revenue and cost Use technology to graph the functions C R and P on the same grid How many booklets must be sold to turn a profit

Determine whether the lines L and L are parallel skew or intersecting L X 49t y 9 6t z 5 3t L x 9 6s y 4s z 5 2s O parallel O skew O intersecting If they intersect find the point of intersection If an answer does not exist enter DNE x y z

7 lim 3 X 0 3 2 10 2 X 00 X 3 limi x 1 2x 8 lim 12x 8 3 X 0 8 x 2x 7

Calculate the number of possibilities Checking out four library books from a list of 8 books for a research project

13 10pts Find the intercepts and asymptotes and then sketch a graph of rational function f x 3x 6 x 4 x 2

2 Without using a calculator if sin 3 with in QIII find the exact value of sin 2

Without using a calculator find cos 20 if csc 0 4 with 0 in QIII

whats a formula for a point that passes through 6 11 0 0 11 0

26 Graph the equation y 2 cos 1 X 4

hat is the value of the absolute maximum of the function in the graph shown below 10 4 7 5 f 2 11 10 0 A 2 3 8 9 10

etermine the interval s for which the function shown below is increasing 8 7 6 5 3 2 the 3 TR 7 5 4 3 19 in 6 10 1 2 3 45678

Given the following piecewise function evaluate f 4 Provide your answer below 3x 1 f x x 1 2x X 3 3 x 0 0 x

3 Find the limit L Then find such that f x L whenever 0 x cl d for e 01 lim 6

Find the value of the limit without a table or graph 1 4 lim 5 8 x 3 x 3 x 5 1 5 lim X 6 X 6 6 lim x 0 cosxtanx X

A baseball card is worth 50 and its value increases at a rate of 23 5 per year

Let f x x 2 x 2 4 Select all that apply fhas a nonremovable discontinuity at x 2 fhas a removable discontinuity at x 2 The limit of fas x approaches 2 does not exist The limit of fas x approaches 2 is 1 4 fhas a nonremovable discontinuity at x 2 fhas a removable discontinuity at x 2

Determine the point s if any at which the graph of the function has a horizontal tangent line If the function has no horizontal tangent line enter NONE y x 1 x y

Use the given information to find f 2 h 2 3 g 2 4 and g 2 2 h 2 1 and f x g x h x f 2

Describe the x values at which the function is differentiable Enter your answer using interval notation f x x 4 2 3 8 6 2 4 5 3 2 1 1 y

Consider the foll 1 x 16 Complete the following table Round your answers to two decimal places 4 1 4 001 f x X 47144 4 5 f x 0 2353 lim f x 1 2346 4 01 12 4844 125 00 X 4 3 999 125 00 X 3 99 12 5156 3 9 3 5 Use the table to determine whether f x approaches or as x approaches 4 from the left and from the right Use a graphing utility to graph the function to c lim f x X 4 0 7900 X 0 2667

According to the theory of relativity the mass m of a particle depends on its velocity v Consider the following equation where mo is the mass when the particle is at rest and c is the speed of light mo 1 1 v c Find the limit of the mass as v approaches c from the left m

Solve the following compound inequality 6x 1 8x 7 6x 9 Submit your answer by dragging the movable blue points on the number line For an interval that extends to or move the blue points to the right or left end of the number line shown as a dashed line You can select whether each endpoint of your chosen interval is open or closed by sliding the respective orange switches up or down Note that the blue line segment above the number line indicates the interval you have chosen and whether the endpoints are open or closed Provide your answer below

Find the one sided limit if it exists If the limit does not exist enter DNE 2 x lim x CSC X 2 x sin x

Consider the following function 1 x 16 Complete the following table Round your answers to two decimal places 4 001 4 3 999 f x X 4 X f x 4 5 lim f x X 4 4 1 4 01 3 99 3 9 Use the table to determine whether f x approaches or as x approaches 4 from the left and from the right Use a graphing utility to graph the function to confirm yo lim f x 3 5

Step 3 Recall the quotient law which states that if the limits lim fx and lim g x exist and lim g x 0 then the following holds x x g x lim f x lim g x X1R Applying this limit law gives us the following 3 00 lim 8 lim 3 HIR X 3

ince x is the highest power of x in the denominator we have determined that we will divide both the numerator and e denominator by x Doing so and simplifying the result gives the following 4x 9 im 3x x 3 lim x 3x x 3 x lim 4x 9 x2 X 3 1 9 4 2 2

Find the quotient 36a 2 29ab 20b 2 4a 5b Enter the correct answer

Graphs of two functions y f x in blue and y g x in red are shown Explain does not exist 3 y f x 2 1 1 lim f x x 2 2 3 4

Suppose hothor lim f r exists or does not exist f x x 3 9 x X 1 if x 3 if x 3

Suppose Fxplain with giving the reason whether lim f x exists or does not exist f x x 3 9 x X if x 3 if x 3

7 Find the limit lim x x 8 x 2 a 8 Process Explanation b 11 og or goizu alimil sr sisulave 3 A16dmur c 2 d 0

Split the angle Of find the fos Set up the sum and difference by using Of 120 the sum and difference formula

Solve 4 cos x 17 cos x 15 0 for all solutions 0 x 2 X Give your answers as values accurate to at least two decimal places in a list separated by commas

Define thermal energy O energy associated with the temperature of an object O energy associated with the motion of an object energy associated with the force of an object O energy associated with the gravity of an object O energy associated with the position or composition of an object

Answer 2 Points 10 Domain 5 Range 10 Key Selecting a button will replace the entered answer value s with the button value If the button is not selected the entered answer is use 10 All real numbers

Simplify 5u v 20u6v Assume that all variables represent positive real numbers 0 63

ind the derivative of the function f t log In 1 1

Find the measure of x 48 9 138 X x Round your answer to the nearest tent

7 Binomial Expansions Expand the following binomials 1 x y x y x y 2 x y x y x y x y on not web sinues inon 3 x 2y 4 x y of qu bebbe eyevle nolansopis ort ni misi dosa no 142 O E E A y x yx 20 MAJ tae mot dovo ot insigilisoo erti asblvong manoer letmonic ent

Expand completely 15 2n 1 to J to morenaqxs ni u to im

11 Find the exact values of the sine and cosine for the angle hyp a Ad 13 OP 1

Find the measure of x 26 x Round to the nearest hundredth X 78

Here s a graph of a function f x 100 90 80 70 60 50 40 30 884 10 6 8 10 s l

Find the distance between the points 7 6 and 3 0 Give an exact answer in simplest radical form Do not round

Convert the polar coordinate 7 Enter exact values X 7 74 y to Cartesian coordinates

4 10 3 7 Enter your answer as a single reduced fractic Multiply

Find the area of the region bounded by 72 cos 20

4 3 2 3 L 2 3 4 q Which of the following is the equation for the graph shown above Or 1 2 sin 8 Or 3 cos 20 Or 3 cos 30 Or 1 2 cos 0 Or 3 sin 20

1 b r 0 and 0 0 2 2 3 4 4 W Identify the coordinates of the point in polar form based upon the given conditions Use pi for or click the yellow arrow for the MathQuill a r 0 and 0 0 2 Details

Jse the precise definition of limits 8 to prove that lim 3 x 5 when x 10