Limits. Calculus Module C06. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

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1 e Calculus Module C Limits Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED March,

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3 Introduction to Limits Statement of Prerequisite Skills Complete all previous TLM modules before beginning this module. Required Supporting Materials Access to the World Wide Web. Internet Eplorer 5.5 or greater. Macromedia Flash Plaer. Rationale Wh is it important for ou to learn this material? An understanding of its is essential to developing an understanding of the derivative introduced in subsequent modules. Learning Outcome When ou complete this module ou will be able to evaluate its. Learning Objectives. Evaluate simple its algebraicall using it notation.. Evaluate simple its graphicall.. Evaluate special its.. Evaluate its b simplifing. 5. Evaluate its in the indeterminate form b simplifing. Connection Activit Suppose ou wish to bu a horse. The owner has set a price of $, but ou hope to get it for less. Inwardl ou decide that each time ou make a new offer, ou will meet the owner halfwa, that is, ou will split the difference. The owner has decided he will never sell for less than $. Your first offer is $, but this is refused. Your second offer is halfwa from $ to $. Then our second offer is $, but the owner sticks to his original price of $. Sticking to our rule that each new offer ou make will represent an increase of one half the difference between our previous offer and the owner s asking price, our successive offers will be $, $, $5, $, $, $, $5, $7.5, $.75; and so on indefinitel. In this eample, our offer is a variable quantit that is alwas approaching $ but NEVER reaches $. We sa that is the it of the variable which is our offer. Note also that the difference between our offer and the it,, becomes smaller and smaller and approaches but never reaches zero. Module C Limits

4 Introduction to Limits The concept of a it is a basic concept in calculus. Some illustrations of its are presented in this section. LIMIT NOTATION Returning to our eample of horse trading, if we represent our offer (the variable) b the letter, then we sa that this variable approaches the constant as a it in such a wa that the difference between the variable and the constant can be made as small as we wish but not zero. Definition: If a variable approaches a constant k in such a wa that the difference between the variable and the constant can be made as small as we wish but not zero, then the constant k is called the it of the variable. The concept of approaching is indicated b an arrow. In words we write: the it of is k. In smbols we write: k. The smbol is used to mean approaches as a it. Thus the epression k is read approaches k as a it. k is called the it of the variable. From the foregoing eample of horse trading, we can write: which is read approaches as a it. Module C Limits

5 OBJECTIVE ONE When ou complete this objective ou will be able to Evaluate simple its algebraicall using it notation. Eploration Activit LIMIT OF A FUNCTION In evaluating its, it is useful to denote a it using mathematical shorthand. The epression k ma be written in more detail in the following manner: = k k This epression is read the it of is k as approaches k. Often we are required to find the it of a function as the independent variable approaches some constant. This is written mathematicall: f ( ) = L k and is read the it of f ( ) as approaches k is L. Definition: The it of f ( ) as approaches k is the value L which is approached b f ( ) as the difference between and k becomes ver small. To understand the meaning of the it of a function, let us take some specific function of. Suppose we take the function. Now let us see what happens to this function as approaches some constant, sa. B the definition of a it, we mean that ma get as close to as we wish but NEVER reach it. The problem ma be written smbolicall: =? Let us select some values of that are successivel closer to, compute the corresponding values of the function, then displa our results in a table of values. We will approach from above (that is, through values greater than the it ), and from below (that is, through values less than the it ). To denote that approaches from above, + we write, with the positive sign to the right of the it. To denote that approaches from below, we write. The sign to the right of the it indicates the direction from which the it is being approached, not the sign of the it. Module C Limits

6 The following tables shows approaching through values greater than : + if = then = ? The following table showing approaching through values less than : if = then = ? Now it appears that as moves closer to, whether from above or from below, the function value moves closer to (a number sandwiched between. and.99 ). In fact, in this instance, if is eactl, the value of the function is eactl. We are interested in what happens to the function as approaches, NOT when =. Our investigation ields an answer to our original problem and we now write the solution in mathematical shorthand: = Our eample ields a useful method of evaluating the it of a function, namel substitution. Simpl stated: As approaches a the function f () approaches the it f (a). ( ) = ( ) f f a a For eample, for the function f( ) = + 5+, as approaches the it, the function approaches the it f () = =. This method of finding its b substitution directl into f () can be used for man algebraic epressions. It is a ver useful and important procedure. It is also fast. EXAMPLE. ( ) ( ) ( ) + + = + + = + + = = = = + = 5 =. 7. ( )( ) ( )( ) ( )( ) = 9 = 7+ = = We shall see later in this module that simple substitution of this kind is NOT alwas possible in evaluating its. NOTE: At this point ou are finding its of algebraic epressions. Module C Limits

7 Eperiential Activit One Evaluate the following its.. 5. ( + ) ( ).. ( 5) +. ( + )( ) ( + ) Eperiential Activit One Answers Module C Limits 5

8 OBJECTIVE TWO When ou complete this objective ou will be able to Evaluate simple its graphicall. Eploration Activit The meaning of the it of a function ma be shown graphicall. Let us sketch the graph of the function = f( ) = as Figure. = Figure (,) + Now let us suppose that approaches some constant from above or below in such a wa that differs from b some small value approaching zero. We note that as, the value approaches, a conclusion alread reached through substitution. Module C Limits

9 EXAMPLE Let f( ) = + 5. Find f ( ). SOLUTION: B substitution, ( + 5) = [( ) + 5] = + 5 = therefore, f( ) = B graphing, we get: 7 5 f ()= + (,) + NOTE: As the value approaches, the value approaches. Therefore, ( + 5) = Module C Limits 7

10 EXAMPLE Find ( + ) SOLUTION: The student is to complete this eample. B substitution,( do our work here) B graphing the it, we get, 7 5 Module C Limits

11 Eperiential Activit Two. Evaluate ( + ) graphicall. Construct a table of values as approaches from above and from below.. Evaluate graphicall. Construct a table of values as approaches from above and from below. ** Wh does the solution b substitution not work with this question?? Eperiential Activit Two Answers. f ( ) = + = f ( ) = f( ) = 5. f( ) = = f ( ) = f( ) = Note: The function is indeterminate at so we cannot substitute to find the it. Module C Limits 9

12 OBJECTIVE THREE When ou complete this objective ou will be able to Evaluate special its. Eploration Activit SOME SPECIAL CASES It is possible for a variable to change in such a manner that it increases or decreases without approaching a it. In this case, we sa the variable increases or decreases without it or without bound. This is written: and is read increases without it, or increases without bound, or becomes infinitel large. It is never correct to sa approaches infinit because infinit cannot be approached. Infinit is NOT a constant and therefore cannot be approached. EXAMPLE Evaluate the following it: (5 7) SOLUTION: In this case if we substitute for in the epression (5 7), then (5 7) approaches infinitel large values, thus (5 7) =, i.e. it gets larger without bound. NOTE: If there is no value L which the epression approaches as the variable changes, we sa the epression (or function) has no it. Module C Limits

13 EXAMPLE = = not defined. Note: You can see that increases without bound as from the positive side but decreases without bound as as from the negative side. The it cannot approach + and at the same time, therefore it does not eist. EXAMPLE EXAMPLE = = = not defined EXAMPLE 5 =, because increases without bound = = = not defined Module C Limits

14 Eperiential Activit Three Evaluate the following its:. ( ). ( + )( ) ( ) ( + ) 7.. ( + )( ) + Eperiential Activit Three Answers Not defined. Not defined NOTE: For computer based (TLM) it questions when the answer is it is entered as.7 Module C Limits

15 OBJECTIVE FOUR When ou complete this objective ou will be able to Evaluate its b simplifing. Eploration Activit In some cases the epression requires algebraic simplification before substitution in order to obtain a real defined it. Our substitution ma lead us to believe there is no it, whereas with a little algebraic manipulation of the epression, a it is easil determined. See the following eamples: EXAMPLE Find the following it if it eists: 9 SOLUTION: Here substitution of = into the epression results in the fraction. We ma, at first glance, want to sa =, but is indeterminate. We ma now be inclined to guess that the it is not defined. This, too, is in error. If we approach from above and below and form a table of values, we find the function does have a it, and that it is. We note that the replacement for in the numerator which produces zero, is the same replacement for in the denominator which ields a zero. This suggests that both numerator and denominator have a common factor. Let us factor the numerator and denominator and thus epress the it in another form: 9 ( + )( ) = ( ) = ( + ) = + = Though the original function is not defined at =, it does have the it. This is important enough to repeat for emphasis: Even though a function ma be undefined at some value of, the it ma still eist. Module C Limits

16 EXAMPLE = ( ) ( )( + ) = = + 7 The division b ( ) before evaluating for the it b substituting = avoids division b zero. This is a common technique for avoiding division b zero. EXAMPLE 5 + ( + )( 5) = + = ( 5) = EXAMPLE = + ( )( ) ( ) ( ) = = NOTE: Even though each of the algebraic epressions in Eamples,, and are undefined for the indicated it on, each eample does have a iting function value. Module C Limits

17 Eperiential Activit Four Evaluate the following its:.. ( ) Eperiential Activit Four Answers. not defined.. not defined. not defined 5. not defined not defined.. not defined 5. The indeterminate form ma also appear in other forms. See Eample. Module C Limits 5

18 OBJECTIVE FIVE When ou complete this objective ou will be able to Evaluate its in the indeterminate form b simplifing. Eploration Activit In some cases the epression requires algebraic simplification before substitution in order to obtain a real defined it. Our substitution ma lead us to believe there is no it, whereas with a little algebraic manipulation of the epression, a it is easil determined. See the following eamples: EXAMPLE Find the it if it eists of the function: SOLUTION: In this function as increases without bound so do both the numerator and demominator increase without bound. i.e. numerator = ( ) + ( ) + 7 =, and denominator = ( ) + ( ) + =. We ma indicate the it as:, which might give the impression that the it = =. However this is not true. Infinit infinit is called an indeterminate form and is a signal that a it might eist and perhaps be found b performing some algebraic operation on the original function. In these tpes we divide both numerator and denominator b to the highest power it occurs in the epression, i.e. b in this eample. We get: = ; NOTE:, 7,, and all go to as. = Thus, the it eists and equals. Module C Limits

19 A Ver Special Limit: e e= ( + ) =. 77 approimatel. The proof that this it eists and has the value indicated is proven in more advanced tets. Here we shall be content with a brief eplanation. Writing u = ( + ) and calculating u when =,. 5,.,.,.,.,., i.e. as we get the following table: u ( + ) ( +. 5) (. ) (. ) (. ) (. ) (. ) = =.5 =.59 =.75 =.79 =.7 =.7 Comparison of the last value of u with e from () above shows that when =., the corresponding value of u differs from e mainl b two units in the fifth decimal place. You ma wish to evaluate u for even smaller values of and see how u compares to e. For instance, let =. and we get u =.75 (approimatel). This inductive approach to calculating u shows that the values for u approach a definite it which is called e. Calculations using e as a base are common in our future stud of mathematics and its applications. Hint: For our mental checks on calculations involving our calculator and base e, remember that e is a value slightl less than. Module C Limits 7

20 Eperiential Activit Five Determine each of the following its: Hint: Divide b Eperiential Activit Five Answers Module C Limits

21 Practical Application Activit Complete the Limits assignment in TLM. Summar The concept of a it was introduced and tied closel to functional notation. The algebraic notation f ( ) = L was stressed. k Limits of functions ma be evaluated: from a table of values as approaches some constant k from above and below graphicall b substitution, and b algebraic manipulation of the function. Eamples and eercises were presented which provided practice in evaluating its. When evaluating its, we are doing nothing more than finding the value being approached b the curve as the value approaches some arbitraril chosen value k. In the delta process module we shall combine the theor from the Theor of Functions module and this module in defining the derivative. Module C Limits 9

22 Review Eercise. Suppose ou wish to sell a horse. A buer is found and offers ou $. Inwardl ou decide that each time the buer refuses our asking price, ou will meet the buer halfwa (split the difference). The buer is firm, but ou are willing to negotiate. Your first asking price is $. This is refused. Your second asking price is halfwa between $ and the buers offer of $. List, in order, our first ten prices which ou would like to receive for our horse.. Evaluate ( ), a) from a table of values as b) graphicall, and c) b substitution. Evaluate the following its: ( ) + and ( ), n n ( ) ( ) n. ( +. 5 ). 5 n n n + 5. n + 7. n 5 Review Eercise Answers. $,,,,,, 5,.5,.5,. a) 5 9. b) 5. c) not defined Module C Limits

23 LIMITS GRAPHS For TLM Limits

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25 Limits

26 Limits

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