Limits & Continuity Questions and Answers

17 E n 1 1 2 3

Make a table of values 4x 5y 45 No Solution X Help 3 2 1 0 1 2 3 Sketch the graph of the equation y 15 1413 12 11 10 9 8 7 6 5 4 3 2 1 H 15 14 43 12 11 40 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f x The graphs of f x and g z are given above Use them to evaluate each quantity below Write DNE if the limit or value does not exist or if it s infinity g x 1 f g 0 2 lim f x g x z 2 3 lim f g x 20 4 f g 2

Find an equation of the tangent line to the curve y 5 2x 3x at 1 0 V

The limit 81 h 9 h lim h 0 is the standard expression defining the derivative of some function f x at some number a Find f and a f x a

Find the limit 19 lim 8 X 8 1 x2 A x2 7x 3 x x3 6x2 4 20 lim A 00 B 8 B 314 C 1 C 1 8 1 D D 0

In 2003 the first complete human genome was sequenced Based on data from the National Human Genome Research Institute 4 the cost of sequencing a human genome in January 2004 was roughly 29 million For the next few years the cost of sequencing a human genome fell by roughly 15 every 6 months a Using this information write a function E t for the cost to sequence a human genome in millions of dollars t years after January 2004 b Using just the information in the problem and not your answer to a what is E 0 What is E 0 5 Check that the function you wrote in a agrees with this If this exponential trend had continued indefinitely 5 in what year would you expect the price of sequencing a human genome to fall below 1 000 6 Please give an exact answer and then use a calculator to give a decimal approximation for this value

d log 5 e log3 y 5 f log 4x 5 d e if y X log then X by

Rewrite using only positive rational exponents 5x 3yz 4 3 5x 3yz Type a single expression raised to a positive expon

Rewrite without rational exponents and simplify if possible 1 6 Simplify your answer Type an exact answer using radicals as nee

Rewrite 64 64 1 with a positive exponent then simplify

Given f x x 7 find an expression that gives the slope of the line from 2 2 to 2 h f 2 h Your answer will be an expression involving h

Let x 2 2 A Sketch the graph of f and when you re done place a 1 in the box values B Find the value of a where f is discontinuous If there is no value enter NONE x 0 x 0 values C Find the value of a where f is not differentiable If there is no value enter NONE

Find the following for the function f x 4x 3x 4 b f 5 c f 5 a f 0 e f x f f x 3 g f 5x a f 0 4 Simplify your answer b f 5 111 Simplify your answer c f 5 81 Simplify your answer 2 d f x 4x 3x 4 Simplify your answer 2 e f x 4x 3x 4 Simplify your answer f f x 3 Simplify your answer d f x h f x

Find the following for the function f x a f 0 e f x X b f 2 f f x 2 2 c f 2 5 Simplify your answer Type an integer or a simplified fraction x 1 X f f x 2 c f 2 g f 4x d f x 2 x 1 Simplify your answer Use integers or fractions for any numbers in the expression d f x h f x h e f x 2 X 1 Simplify your answer Use integers or fractions for any numbers in the expression x 2 x 4 17 Simplify your answer Ilse integers or fractions for any numbers in the expression

4 In 3 4 sentences describe the main similarities and differences between prevalence and incidence Consider both what information goes into the formula but also how or why that information is important and could be used

14 lim h 0 x h 3 x3 h A 3x2 3xh h B Does not exist C 3x2

Find the limit if it exists x2 100 X 10 13 lim X 10 A 1 B 20 C 10 D Does not exist

Evaluate the following limits 2 lim I 3 I 3 1 1 lim 17 x x 7 2 lim 2 3 S 3 1 lim x 0 x x 7 DNE X DNE

Evaluate the following limits If needed enter INF for and INF for a b lim I 00 373 37 11x 7 8x 5x3 3x3 3x 11x lim 2 00 7 8x 5x3

Evaluate the following limits If needed enter INF for o and INF for b 10x 6 1lz 200 3 6x 9 11x T 10x 6 1lz JUNE

A function is said to have a vertical asymptote wherever the limit on the left or right or both is either positive or negative infinity For example the function f x 2 11 has a vertical asymptote at 5 213 x 5 For each of the following limits enter either P for positive infinity N for negative infinity or D when the limit simply does not exist x 1 lim 25 2x 3 x 5 1 I5 2x 3 x x 1 lim 25 2x 3 x 5 5

Let f x 7 x 1 x 3 Use interval notation to indicate the largest set where f is continuous Largest set of continuity 1 0 T

Evaluate the following limits If needed enter INF for oo and INF for a b lim I lim 2 4x 8 5 2 4x

Describe the set of a values shown below using interval and algebraic notation 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8

Sketch the graph of the following function and use it to answer the following questions 4 x if a 2 Let f x if 2 x 3 if x 3 x 3 Calculate the following limits Enter DNE if the limit does not exists lim f x TORRENT lim f x I 2 lim f x I 2 lim f x lim f x I 3 lim f x I 3

Let f x x 8 Use interval notation to indicate the largest set where f is continuous Largest set of continuity

118 Evaluate each of the following expressions Note Enter DNE if the limit does not exist or is not defined a lim F x I 1 b lim F x c lim F x I 1 d F 1 e lim F x z 1 f lim F x I 1 g lim F x

For what value of the constant c is the function f defined below continuous on if x 0 8 if z 8 00 f x H T C CI 3 PE

Instructions Show your work to receive full credit on each problem you do not need to show every arithmetic step Use algebraic techniques to solve problems unless otherwise instructed Simplify and circle your final answers 1 10 pts Find the intercepts of the equation y Name x 7x 2 x 3

Let f be a continuous function such that f 5 1 and f 5 1 Using the Intermediate Value Theorem classify the following statements as A Always true B Never True or C True in some cases False in others 1 f 0 0 Answer A B or C 2 For some c where 5 c 5 c 0 Answer A B or C

6 x 5 Find each point of discontinuity of f and for each give the value of the point of discontinuity and evaluate the indicated one sided limits NOTE Use INF for o and INF for If you have more than one point give them in numerical order from smallest to largest If you have extra boxes fill each in with an x Point 1 C lim f x I C lim f x I CI Point 2 C lim f x I C lim f x I C Point 3 C lim f x I C lim f x f x

determine lim f x I 2 What theorem did you use to arrive at your answer 5x 14 f x x x 10

Let lim f x 0 lim h x 2 lim g x 5 I a I a 20 Find following limits if they exist If not enter DNE does not exist as your answer 1 lim f x h x I X 2 lim f x h x I a 3 lim f x g x IX f x Ia h x 4 lim 5 lim I Ta g x g x 6 lim Isa Ta f x 7 lim h x xa 8 lim h x 1 Ta 1 za h r g x I 9 lim

A function f x is said to have a jump discontinuity at x a if 1 lim f x exists I a 2 lim f x exists I a 3 The left and right limits are not equal Let f x Show that f x has a jump discontinuity at x 10 by calculating the limits from the left and right at x 10 lim f x 210 lim f x x 10 H 5x 2 if a 10 4 if a 10 x 5 SARE Now for fun try to graph f x

The answer above is NOT correct Find in terms of the constant a lim h 0 4 a h h V 4a

Let 1 25 x 5 Sketch the graph of this function and find the following limits if they exist if not enter DNE 1 lim F x I 5 2 lim F x I5 3 lim F x

find the domain and range of the function 1 f x 3 x A D R 3 00 C D 0 R 0 3 8 2 1 A D 0 R 00 00 C D 0 1 R 0 0 2 g z B D 0 0 R 3 0 D D 0 R 0 B D 1 0 R 00 00 D D 1 0 R 0

4 Perform the following operations Assume all numbers are measured unless otherwise indicated Give all answers in scientific notation and with the correct number of significant figures 4 points a 7 83 x 10 5 5 x 10 3 915 108 b 4 6 x 108 8 38 x 197 5 5 48 92601 431 98 102

The graph below is the function f x 5 4 3 2 1 3 2 1 2 4 5 Determine which one of the following rules for continuity is violated first at x 2 Of a is defined Olim f x exists xia Olim f x f a HIP

The graph below is the function f x 5 4 3 2 1 5 4 3 2 1 1 N 3 4 5 ON Select all statements below that you agree with Note You may be checking more than one box No partial credi Of 2 is defined lim f x exists x 2 4 5 Olim f x f 2 x 2 The function is continuous at x 2 The function is not continuous at x

A 5 2 resistor and a variable resistor of resistance R are placed in parallel The expression for the resulting resistance R is given by R Determine the limiting value of R as R The limiting value of R is 1 Simplify your answer 5R 5 R

27 32 Graphing inverse functions Find the inverse function on the given interval if specified and graph both f and f on the same set of axes Check your work by looking for the required symmetry in the graphs 27 f x 8 4x 29 f x x 2 for x 2 30 f x 3 x for x 3 31 f x x 2 1 for x 2 32 f x x 4 for x 0 28 f x 3x 5

Parking fees Suppose that it costs 5 per minute to park at the air port with the rate dropping to 3 per minute after 9 P M Find and graph the cost function c t for values of t satisfying 0 t 120 Assume that t is the number of minutes after 8 P M

Below is the graph of y x Transform it to make it the graph of y 3 x 1 4 VN X

The box plot below represents some data set What is the range of the data T

Find the surface area of a square pyramid with side length 3 m and slant height 4 m 2 m m 3 m

The graph below represents data that was collected by students on the number of social media posts they made the previous day SOCIAL MEDIA POSTS Frequency Number of Students 1 2 3 4 5 6 7 Number of Social Media Posts Made 8 9 What s the most common number of posts students in this class made

Evaluate the function at the specified points h x y z xy 4 3 2 3 2 2 At 4 3 2 At 3 2 2

7 Kalia is planning the transportation for the senior trip The number of students in the senior class is 463 but the trip is entirely voluntary If each bus can seat 48 students describe the set of the number of busses b they may need in set notation