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answers on a separate machinrable answer sheet Total testing time is two hours and fi fty minutes there are no separately timed sections Following are some general testtaking strategies you may want to consider ˜ Read the test directions carefully and work as
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x2 x 1 19 2 using the de nition of a limit 1 The rules of the game Normally the answer to this kind of question will be of the following form Given 0 let something positive usually depending on and a If 0 jx ajthen some series of steps goes here so jfx Lj
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Answer True All polynomial functions are continuous functions and therefore lim px as x approaches a pa Question 8 True or False If lim fx L1 as x approaches a from the left and lim fx L2 as x approaches a from the right lim fx as x approaches a exists only if L1 L2 Answer True This is an important property of the limits
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Answers to these questions are also presented Questions and Answers on Limits in Calculus A set of questions on the concepts of the limit of a function in calculus are presented along with their answers Questions and Answers on Continuity of Functions Questions on the concepts of continuity and continuous functions in calculus are presented
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If the function for which the limit needs to be computed is defined by an algebraic expression which takes a finite value at the limit point then this finite value is the limit value 3 If the function for which the limit needs to be computed cannot be evaluated at the limit point ie the value is an
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Problems sm l lim x lim rcscx 2 10 sm 3r lim 6 x0 sin 2x sin 3x lim 10 5x sm ax lim sinbx lim 18 x0 COS X 3 7 11 15 19 lim lim sm 2x sm3x sm 4x sm ax lim 4 smx lim 8 2x 3x lim 12 x0 Slll X sin 5x lim 16 5x I cosx lim 20 tan x 5 lim 3smx 9 lim sm x 13 lim 10 I cos2x 17 lim x0 2x Answers lim x0 2x sin
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Limits and Continuity MULTIPLE CHOICE Choose the one alternative that best completes the statement or answers the question Solve the problem 1Assume that a watermelon dropped from a tall building falls y 16t2 ft in t sec Find the watermelons average speed during the first 6 sec of fall A97 ftsec B48 ftsec C96 ftsec D192 ftsec 1
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Answers 1 1 2 05 3 infinity 4 no limit 5 01353 6 05774 7 x4x2 the limit is 0 8 x3x the limit is ¥ 9 6x2x the limit is 3 10 x2 x2 the limit is 1 11 x23x2 the limit is 13 12 4x3x2 the limit is ¥ 13 x22x the limit is 0 14 3xx5 the limit is ¥ 15 01 16 23 17 025 18 5 19 10 20 3 21 6 22 3
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62 Chapter 2 Limits and Continuity 6 Power Rule If r and s are integers s 0 then lim x→c f x r s Lr s provided that Lr s is a real number The limit of a rational power of a function is that power of the limit of the function provided the latter is a real number THEOREM 2
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tangent line problemand the area problem We will see in this and the subsequent chapters that the solutions to both problems involve the limit concept 67 21 LimitsAn Informal Approach 22 Limit Theorems 23 Continuity 24 Trigonometric Limits 25 Limits That Involve Infinity 26 LimitsA Formal Approach 27 The Tangent Line Problem Chapter 2 in Review
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201103RE Calculus 1 WORKSHEET LIMITS 1 Use the graph of the function fx to answer each question Use 1 1 or DNEwhere appropriate a f0 b f2 c f3 d lim
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The reason that the limit is 9 is that our new function fx coincides with our old continuous function gx for all xexcept x 3 Therefore the limit of fx as x3 is the same as the limit of gx as x3 and since gis continuous this is g3 9 106 Sandwich in a bow tie We return to the function from example Consider fx xcos 1 x for x6 0
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131 Overview 1311 Limits of a function Let f be a function defined in a domain which we take to be an interval say I We shall study the concept of limit of f at a point ''a'' in I We say – lim x a f x → is the expected value of f at x a given the values of f near to the left of value is called the left hand limit of f at a We say lim
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The week of March 23rd we will be reviewing Limits and Continuity The session will begin in room 315 with a brief review of the weekly topic Instruction will be from 300 pm to 315 pm Once we have reviewed the topic you may begin practicing the questions in your review packet Answers will be posted in room 315 and 312 all week and will be posted on line
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Background Topics limit of fx as xapproaches a limit of fx as xapproaches in nity left and righthand limits 411 De nition Suppose that fis a real valued function of a real variable ais an accumulation point of the domain of f and ''2R
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Beginning Differential Calculus Problems on the limit of a function as x approaches a fixed constant limit of a function as x approaches plus or minus infinity limit of a function using the precise epsilondelta definition of limit limit of a function using lHopitals rule Problems on the continuity of
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Limits and Continuity Answer Keys Problem Set 1 – Tangent Lines 1 2 2 4 3 84 or 4 6 4 5 5
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Limits at Infinity Problems and Solutions To determine which we use our usual approach and look at just the term with the highest power in the numerator and the term with the highest power in the denominator lim x→∞x32 3x24lim x→∞x3 3x2 lim x→∞x 3 ∞✓ The second line in the solution shows that the function approaches x 3 as x grows large
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1 1 If 1 is not finite and sin is undefined 1 1 If 1 is finite and sin is defined a nd also finite Condition for continuity of a function f at a point is limf f A number for which an expression f ei ther is undefined or infinite is called a of the function f
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Limit and Continuity 20 LIMIT AND CONTINUITY Consider the function x12 fx x1 − − You can see that the function fx is not defined at x 1 asx1− is in the denominator Take the value of x very nearly equal to but not equal to 1 as given in the tables below In this case x1− ≠ 0 as x ≠ 1 ∴ We can write x12 x1x1 fxx1 x1x1 − −
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Answer a This problem requires a geometrical argument Solution 1 By similar triangles f x 6 −a 0−a gx 3 and therefore fx gx 6 3 2 Solution 2 lim x→a fx gx lim x→a fx −a gx −a lim x→a slope of f slope of g 6 3 2 This problem is a nice preview of
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Limits and Continuity 21 An Introduction to Limits 22 Properties of Limits 23 Limits and Infinity I Horizontal Asymptotes HAs 24 Limits and Infinity II Vertical Asymptotes VAs 25 The Indeterminate Forms 00 and 26 The Squeeze Sandwich Theorem 27 Precise Definitions of Limits 28 Continuity
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AP Calculus Limits and Continuity AP Calculus Learning Objectives Explored in this Section Express limits symbolically using correct notation Interpret limits expressed symbolically Estimate limits to functions Answers in the back Limits at Jump Discontinuities and Kinks
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