Limits & Continuity Questions and Answers

Find lim P 4 5 2 3 1 x 5 y 1 2 3 X 5 1 lim x 5 y 1 2 3 0 X 5 P 4 5 2 Type an integer or a simplified fraction

Find the smallest number n of terms needed to obtain an approximation of the series 10 4 n 8 k6 k 1 accurate to

For the function f whose graph is given state the following If the limit is infinite enter or as y a b c d 6 lim f x 0 00 lim f x 8118 3 x 4 X 2 2 2 2 6

Recall sum law which states that if the limits lim f x and lim g x exist then the following holds x 00 x40 lim f x g x g x lim f x lim g x 818 818 The difference law is similar to the above but deals with a difference of functions as opposed to a sum f x g x lim f x lim g x x 0 lim 5 X18 lim 818 Applying these limit laws gives us the following Jim 8 37 2 lim 8 lim X 8 X18 lim 5 lim 818 lim X X X

tep 2 Since x is the highest power of x in the denominator we have determined that we will divide both the numerator and the denominator by x2 Doing so and simplifying the result gives the following 8x 7 x 5x2 x 3 lim 8x 7 lim x o 5x2 x 3 x lim x18 5 8

How much work is done lifting a 35 pound object from the ground to the top of a 45 foot building if the cable used weighs 0 5 pounds per foot Question Help foot pounds Video Message instructor

Which unit is most appropriate for measuring the length of a beetle that is about the size of your fingertip A kilometer km B centimeter cm C decimeter dm D decameter dam

Find an equation for a negative cosine function having an amplitude of 3 a period of 2 and a horizontal shift of 2 to the left 12 y cos x

Find the horizontal shift y cos 5x 2TT T Enter either or Ente

Express the number in standard notation or standard form 5 74 108 X Standard Form Enter

Find the period of this function y 2 sin 40

Use the graph to estimate the open intervals on which the function is increasing or decreasing Then find the open intervals analyticall Enter your answers using interval notation y x 1 increasing decreasing 3 2 2 3 X

Find each limit if it exists X x 7 lim x x 6 a b c x 7 lim X x4 6 x 7 lim X x 6 X

Question 3 Evaluate the limit using L H pital s rule 10 11 lim I 0 7 X

Find each x value at which f is discontinuous and for each x value determine whether f is continuous from the right or from the left or neither if x 0 x 7 ex 3 x if 0 x 1 if x 1 X f x smaller value Is f continuous from the right left or neither at this value O continuous from the right O continuous from the left O neither X larger value Is f continuous from the right left or neither at this value O continuous from the right O continuous from the left O neither Sketch the graph of f y 10 5 y 10 5 y 10 j 5

its sides A D b A O c As a single vector AF DB is O a CD BF E O d EB B

Express in the form a bi 1 i O 1 i 1

11 a lim sin 2x X EN 2 b lim tan X 3

8 Use the graph to answer the questions Ma PWS jung on kas w PMK 2 Kinsh M H AS 4 a Give the coordinates of the point P b Find the y coordinate for the point on the graph where x 3 c Find the x coordinate for the point on the graph where y 5 d Find the y intercept of the graph

Which equation results after clearing the radical from the equation x 5 3 X Select an answer and submit For keyboard navigation use the up down arrow keys to select an answer TO x 5 9 x b x 5 x 9 C x 5 x 6x 9 d n G

Given that lim 4x 6 2 illustrate this definition by finding the largest values of that correspond to 0 5 0 1 and 0 05 x 2 8 s 0 1 s 8 s 0 5 0 05

Find the limit If the limit is infinite enter o or o as appropriate If the limit does not otherwise exist enter DNE 7x5 2 X lim xx 6

SOLUTION 1 Preliminary analysis of the problem guessing a value for 6 Let be a given positive number We want to find a number such that if 0 x 4 8 then 3x 10 2 E But 3x10 2 3x 12 3 if 0 x 41 6 then 3 that is if 0 x 41 6 then This suggests that we should choose 3 2 Proof showing that works Given 0 choose If 0 If 3 3x10 2 3 3 E E Therefore we want such that 36 Thus if 0 x 4 6 then 3x 10 21 E Therefore by the definition of a limit we get the following lim 3x 10 2 x 4 E 23 6 then we get the following

A C Which of the following represents y cot x T B D 4

A C Which of the following represents y sec x V V U U 2 n t n B D VEN TAT UNI in

Which function is periodic A C AAA w B D MAL

Find the limit If the limit is infinite enter or as appropriate If the limit does not otherwise exist enter DNE V1 25x6 6 x3 lim X18

Find the indicated power and provide your answer in standard form a bi cos i sin B i D i A i C i 2 Enter

Determine the location of any critical values of f If multiple such values exist enter the solutions using a comma separated list 02 N 4 4 5 5 5 5 6 6 5 7 7 5 8 8 5 9 9 5 10 9 5 IC No critical values exist Determine the location of any absolute minima of f If multiple such values exist Inter the solutions using a comma separated list OX No absolute minima exist Determine the location of any absolute maxima of f If multiple such values exist enter the solutions using a comma separated list No absolute maxima exist etermine the location of any relative minima of f If multiple such values exist enter the solutions using a omma separated list X No relative minima exist termine the location of any relative maxima of f If multiple such values exist enter the solutions using a

7 The function f x x and its inverse f x x Find their derivatives and verify the theorem f x 1 f f x 0 f f x

ind the equation of the quadratic function f whose graph is shown below 12 10 D 4 2 6 10 2 f x

11 12 Sketch the graph of the function and use it to determine the values of a for which limx a f x exists 11 f x x 1 In x if x 0 if 0 x 1 if x 1 Any help would be much appreciated

2x 7 x 4 16 For what value of k is g x 3x k x 4 function A 0 B 4 C 4 D 27 E 27 a continuous

Use continuity to evaluate the limit lim x 34 x X 5

Evaluate the limit using the appropriate properties of limits 8x 7 x 5x2 x 3 lim Step 1 We are given the following 8x 7 lim x o 5x2 x 3 We note that as x becomes large both the numerator and the denominator become large Therefore it is not immediately obvious what happens to their ratio So we need to do s preliminary algebra Recall that to evaluate the limit of a rational function at infinity we first divide both the numerator and the denominator by the highest power of x that occurs denominator Therefore we should divide the numerator and the denominator by which of the following Ox O x

11 Find the limit lim o A 1 B 0 C D 1 2 tan y cos y y

Find the measure of x 45 X 600 Round to the nearest tenth X En

Find B in degrees B 14 15 70 b C B degrees Round to the nearest tenth

What is the area of this sector 3 75 4 T sq units Enter

Consider the following function f x X 5 x 25 a Explain why f has a removable discontinuity at x 5 Select all that apply lim f x is finite X 5 lim f x does not exists X 5 f 5 is undefined f 5 and lim f x are finite but are not equal X 5 O none of the above b Redefine f 5 so that f is continuous at x 5 and thus the discontinuity is removed f 5

8 Find the limit lim A 0 B 011214 3 7 8 2t C 1 D 4 E The limit does not exist

What is cos 11 110 Enter or Ent

7 Find the limit lim A 0 B 1 C 1

1 What is lim A 1 B 0 C 1 D E 2 2 3 tan

18 Which of the following are point s of discontinuity of the 3x 1 function f x 2x 8x 64x I O II 4 III 8 A I only B II only C III only D I and II only

4x 9 Find the limit lim 0 1 cos 2x A 0 B 2 C 4 D 16 E The limit does not exist

4 Find the limit 3 o A B 8 14 4 1 2 lim 0 t C 0 1 D 4 E The limit does not exist

Question 9 Integrate 3 cot 3 dx f3 cot

Divide 8x 9x 12x 13 3x 4x 1 Write your answer in the following form Quotient 3 8x 9x 12x4 13 3x 2 4x 1 CO 7 N Remainder 4x 1

Guess the value of the limit lim X18 by evaluating the function f x BR for x 0 1 2 3 4 5 6 7 8 9 10 20 50 and 100 Use a graph of f to support y 5