Properties of Limits
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1 33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate a it using dividing out and rationalizing techniques Evaluate a it using the Squeeze Theorem Properties of Limits In Section, ou learned that the it of f as approaches c does not depend on the value of f at c It ma happen, however, that the it is precisel fc In such cases, the it can be evaluated b direct substitution That is, f fc Substitute c for Such well-behaved functions are continuous at c You will eamine this concept more closel in Section 4 f(c) = THEOREM Some Basic Limits c + ε f(c) = c ε = δ Let b and c be real numbers and let n be a positive integer b b c 3 n c n ε = δ c ε c δ Figure 6 c c + δ Proof To prove Propert of Theorem, ou need to show that for each > 0 there eists a > 0 such that c < whenever 0 < c < To do this, choose The second inequalit then implies the first, as shown in Figure 6 This completes the proof (Proofs of the other properties of its in this section are listed in Appendi A or are discussed in the eercises) NOTE When ou encounter new notations or smbols in mathematics, be sure ou know how the notations are read For instance, the it in Eample (c) is read as the it of as approaches is 4 EXAMPLE Evaluating Basic Limits a 3 3 b 4 c 4 4 THEOREM Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following its f L and g K Scalar multiple: Sum or difference: 3 Product: bf bl f ± g L ± K fg LK 4 Quotient: f g L K, provided K 0 5 Power: f n L n
2 33460_003qd //04 :3 PM Page CHAPTER Limits and Their Properties EXAMPLE The Limit of a Polnomial Propert Propert Eample Simplif In Eample, note that the it (as ) of the polnomial function p 4 3 is simpl the value of p at p p This direct substitution propert is valid for all polnomial and rational functions with nonzero denominators THEOREM 3 Limits of Polnomial and Rational Functions If p is a polnomial function and c is a real number, then p pc If r is a rational function given b r pq and c is a real number such that qc 0, then pc r rc qc EXAMPLE 3 The Limit of a Rational Function Find the it: Solution Because the denominator is not 0 when, ou can appl Theorem 3 to obtain 4 Polnomial functions and rational functions are two of the three basic tpes of algebraic functions The following theorem deals with the it of the third tpe of algebraic function one that involves a radical See Appendi A for a proof of this theorem THE SQUARE ROOT SYMBOL The first use of a smbol to denote the square root can be traced to the siteenth centur Mathematicians first used the smbol, which had onl two strokes This smbol was chosen because it resembled a lowercase r, to stand for the Latin word radi, meaning root THEOREM 4 The Limit of a Function Involving a Radical Let n be a positive integer The following it is valid for all c if n is odd, and is valid for c > 0 if n is even n c n
3 33460_003qd //04 :3 PM Page 6 SECTION 3 Evaluating Limits Analticall 6 The following theorem greatl epands our abilit to evaluate its because it shows how to analze the it of a composite function See Appendi A for a proof of this theorem THEOREM 5 The Limit of a Composite Function If f and g are functions such that g L and f fl, then fg f g f L L EXAMPLE 4 The Limit of a Composite Function a Because and 0 it follows that b Because and it follows that You have seen that the its of man algebraic functions can be evaluated b direct substitution The si basic trigonometric functions also ehibit this desirable qualit, as shown in the net theorem (presented without proof) THEOREM 6 Limits of Trigonometric Functions Let c be a real number in the domain of the given trigonometric function sin sin c 3 tan tan c 4 cos cos c cot cot c 5 sec sec c 6 csc csc c EXAMPLE 5 Limits of Trigonometric Functions a tan tan0 0 0 b cos cos cos c 0 sin 0 sin 0 0
4 33460_003qd //04 :3 PM Page 6 6 CHAPTER Limits and Their Properties A Strateg for Finding Limits On the previous three pages, ou studied several tpes of functions whose its can be evaluated b direct substitution This knowledge, together with the following theorem, can be used to develop a strateg for finding its A proof of this theorem is given in Appendi A f() = 3 3 THEOREM 7 Functions That Agree at All But One Point Let c be a real number and let f g for all c in an open interval containing c If the it of g as approaches c eists, then the it of f also eists and f g EXAMPLE 6 Finding the Limit of a Function Find the it: 3 3 g() = + + f and g agree at all but one point Figure 7 Solution Let f 3 B factoring and dividing out like factors, ou can rewrite f as f So, for all -values other than, the functions f and g agree, as shown in Figure 7 Because g eists, ou can appl Theorem 7 to conclude that f and g have the same it at 3 3 g, Factor Divide out like factors Appl Theorem 7 Use direct substitution Simplif STUDY TIP When appling this strateg for finding a it, remember that some functions do not have a it (as approaches c) For instance, the following it does not eist 3 A Strateg for Finding Limits Learn to recognize which its can be evaluated b direct substitution (These its are listed in Theorems through 6) If the it of f as approaches c cannot be evaluated b direct substitution, tr to find a function g that agrees with f for all other than c [Choose g such that the it of g can be evaluated b direct substitution] 3 Appl Theorem 7 to conclude analticall that f g gc 4 Use a graph or table to reinforce our conclusion
5 33460_003qd //04 :3 PM Page 63 SECTION 3 Evaluating Limits Analticall 63 Dividing Out and Rationalizing Techniques Two techniques for finding its analticall are shown in Eamples 7 and 8 The first technique involves dividing out common factors, and the second technique involves rationalizing the numerator of a fractional epression EXAMPLE 7 Dividing Out Technique Find the it: ( 3, 5) f is undefined when 3 Figure 8 NOTE In the solution of Eample 7, be sure ou see the usefulness of the Factor Theorem of Algebra This theorem states that if c is a zero of a polnomial function, c is a factor of the polnomial So, if ou appl direct substitution to a rational function and obtain rc pc qc 0 0 ou can conclude that c must be a common factor to both p and q f() = Solution Although ou are taking the it of a rational function, ou cannot appl Theorem 3 because the it of the denominator is Direct substitution fails Because the it of the numerator is also 0, the numerator and denominator have a common factor of 3 So, for all 3, ou can divide out this factor to obtain f 6 3 Using Theorem 7, it follows that g, Appl Theorem Use direct substitution This result is shown graphicall in Figure 8 Note that the graph of the function f coincides with the graph of the function g, ecept that the graph of f has a gap at the point 3, 5 In Eample 7, direct substitution produced the meaningless fractional form 00 An epression such as 00 is called an indeterminate form because ou cannot (from the form alone) determine the it When ou tr to evaluate a it and encounter this form, remember that ou must rewrite the fraction so that the new denominator does not have 0 as its it One wa to do this is to divide out like factors, as shown in Eample 7 A second wa is to rationalize the numerator, as shown in Eample 8 TECHNOLOGY PITFALL Because the graphs of 3 δ 5 + ε 3 + δ f 6 3 and g Incorrect graph of Figure 9 f Glitch near ( 3, 5) 5 ε differ onl at the point 3, 5, a standard graphing utilit setting ma not distinguish clearl between these graphs However, because of the piel configuration and rounding error of a graphing utilit, it ma be possible to find screen settings that distinguish between the graphs Specificall, b repeatedl zooming in near the point 3, 5 on the graph of f, our graphing utilit ma show glitches or irregularities that do not eist on the actual graph (See Figure 9) B changing the screen settings on our graphing utilit ou ma obtain the correct graph of f
6 33460_003qd //04 :3 PM Page CHAPTER Limits and Their Properties EXAMPLE 8 Rationalizing Technique Find the it: 0 Solution B direct substitution, ou obtain the indeterminate form Direct substitution fails 0 0 f() = + In this case, ou can rewrite the fraction b rationalizing the numerator, 0 Now, using Theorem 7, ou can evaluate the it as shown 0 0 The it of f as approaches 0 is Figure 0 A table or a graph can reinforce our conclusion that the it is (See Figure 0) approaches 0 from the left approaches 0 from the right f ? f approaches 05 f approaches 05 NOTE The rationalizing technique for evaluating its is based on multiplication b a convenient form of In Eample 8, the convenient form is
7 33460_003qd //04 :3 PM Page 65 SECTION 3 Evaluating Limits Analticall 65 f g h h() f() g() The Squeeze Theorem Figure f lies in here g c h f The Squeeze Theorem The net theorem concerns the it of a function that is squeezed between two other functions, each of which has the same it at a given -value, as shown in Figure (The proof of this theorem is given in Appendi A) THEOREM 8 The Squeeze Theorem If h f g for all in an open interval containing c, ecept possibl at c itself, and if h L g then f eists and is equal to L You can see the usefulness of the Squeeze Theorem in the proof of Theorem 9 THEOREM 9 Two Special Trigonometric Limits sin cos (cos θ, sin θ) (, tan θ) Proof To avoid the confusion of two different uses of, the proof is presented using the variable, where is an acute positive angle measured in radians Figure shows a circular sector that is squeezed between two triangles FOR FURTHER INFORMATION For more information on the function f sin, see the article The Function sin b William B Gearhart and Harris S Shultz in The College Mathematics Journal To view this article, go to the website wwwmatharticlescom θ (, 0) A circular sector is used to prove Theorem 9 Figure tan θ sin θ θ θ θ Area of triangle Area of sector Area of triangle tan sin Multipling each epression b sin produces cos sin and taking reciprocals and reversing the inequalities ields cos sin Because cos cos and sin sin, ou can conclude that this inequalit is valid for all nonzero in the open interval, Finall, because cos and, ou can appl the Squeeze Theorem to 0 0 conclude that sin The proof of the second it is left as an eercise (see 0 Eercise 0)
8 33460_003qd //04 :3 PM Page CHAPTER Limits and Their Properties EXAMPLE 9 A Limit Involving a Trigonometric Function Find the it: tan 0 f() = tan 4 The it of f as approaches 0 is Figure 3 Solution Direct substitution ields the indeterminate form 00 To solve this problem, ou can write tan as sin cos and obtain tan 0 0 sin cos Now, because sin 0 ou can obtain (See Figure 3) and 0 tan 0 sin 0 0 cos cos EXAMPLE 0 A Limit Involving a Trigonometric Function Find the it: 0 sin 4 g() = sin 4 6 The it of g as approaches 0 is 4 Figure 4 Solution Direct substitution ields the indeterminate form 00 To solve this problem, ou can rewrite the it as sin sin Multipl and divide b 4 Now, b letting 4 and observing that 0 if and onl if 0, ou can write sin sin (See Figure 4) 4 sin 0 4 TECHNOLOGY Use a graphing utilit to confirm the its in the eamples and eercise set For instance, Figures 3 and 4 show the graphs of f tan and g sin 4 Note that the first graph appears to contain the point 0, and the second graph appears to contain the point 0, 4, which lends support to the conclusions obtained in Eamples 9 and 0
9 33460_003qd //04 :3 PM Page 67 SECTION 3 Evaluating Limits Analticall 67 Eercises for Section 3 In Eercises 4, use a graphing utilit to graph the function and visuall estimate the its h 5 g In Eercises 5, find the it h 5 h 3 f cos 4 f f ft t g 4 g f t t t 4 ft t tan In Eercises 37 40, use the information to evaluate the its 37 f 38 (c) (d) 39 f 4 40 (c) (d) See wwwcalcchatcom for worked-out solutions to odd-numbered eercises g 3 f 3f f 3 5g f g f g f g f 3 (c) (d) f 7 3f In Eercises 4 44, use the graph to determine the it visuall (if it eists) Write a simpler function that agrees with the given function at all but one point 4 4 h 3 g 3 sec 7 6 f 3 g (c) (d) f 4f f g f g f g f 8 f 3 3 In Eercises 3 6, find the its 3 f 5, g 3 f g (c) 4 4 f 7, g f g (c) f 4, g f g (c) 3 6 f 3, g 3 6 f g (c) 4 In Eercises 7 36, find the it of the trigonometric function 7 sin 8 tan 9 30 sin 3 3 sec 3 cos sin cos g f g f 3 g f g f 4 g 0 g 3 g g 43 g 3 44 f 3 5 h h 0 f f 0 3
10 33460_003qd //04 :3 PM Page CHAPTER Limits and Their Properties In Eercises 45 48, find the it of the function (if it eists) Write a simpler function that agrees with the given function at all but one point Use a graphing utilit to confirm our result In Eercises 49 6, find the it (if it eists) Graphical, Numerical, and Analtic Analsis In Eercises 63 66, use a graphing utilit to graph the function and estimate the it Use a table to reinforce our conclusion Then find the it b analtic methods In Eercises 67 78, determine the it of the trigonometric function (if it eists) 67 sin sin cos sin cos h h 0 h cos cot sin 3t t 0 t sin 0 sin 3 4 sin Hint: Find cos 0 cos tan 0 tan 0 sec tan sin cos 3 3 sin 3 Graphical, Numerical, and Analtic Analsis In Eercises 79 8, use a graphing utilit to graph the function and estimate the it Use a table to reinforce our conclusion Then find the it b analtic methods sin 3t cos t 0 t sin sin In Eercises 83 86, find 83 f 3 84 f 85 f 4 86 f 4 In Eercises 87 and 88, use the Squeeze Theorem to find f 87 c 0 4 f 4 88 c a b a f b a In Eercises 89 94, use a graphing utilit to graph the given function and the equations and in the same viewing window Using the graphs to observe the Squeeze Theorem visuall, find f 89 f cos 90 f sin 9 f sin 9 f cos 93 f sin 94 h cos 99 Writing Use a graphing utilit to graph f, g sin, and h sin in the same viewing window Compare the magnitudes of f and g when is close to 0 Use the comparison to write a short paragraph eplaining wh h Writing About Concepts f f 95 In the contet of finding its, discuss what is meant b two functions that agree at all but one point 96 Give an eample of two functions that agree at all but one point 97 What is meant b an indeterminate form? 98 In our own words, eplain the Squeeze Theorem
11 33460_003qd //04 :3 PM Page 69 SECTION 3 Evaluating Limits Analticall Writing Use a graphing utilit to graph f, g sin, and h sin in the same viewing window Compare the magnitudes of f and g when is close to 0 Use the comparison to write a short paragraph eplaining wh h 0 0 Free-Falling Object In Eercises 0 and 0, use the position function st 6t 000, which gives the height (in feet) of an object that has fallen for t seconds from a height of 000 feet The velocit at time t a seconds is given b sa st t a a t 0 If a construction worker drops a wrench from a height of 000 feet, how fast will the wrench be falling after 5 seconds? 0 If a construction worker drops a wrench from a height of 000 feet, when will the wrench hit the ground? At what velocit will the wrench impact the ground? Free-Falling Object In Eercises 03 and 04, use the position function st 49t 50, which gives the height (in meters) of an object that has fallen from a height of 50 meters The velocit at time t a seconds is given b sa st t a a t 03 Find the velocit of the object when t 3 04 At what velocit will the object impact the ground? 05 Find two functions f and g such that f and g do 0 0 not eist, but f g does eist 0 06 Prove that if f eists and f g does not eist, then g does not eist 07 Prove Propert of Theorem 08 Prove Propert 3 of Theorem (You ma use Propert 3 of Theorem ) 09 Prove Propert of Theorem 0 Prove that if f 0, then f 0 Prove that if f 0 and for a fied number g M M and all c, then fg 0 Prove that if f 0, then f 0 (Note: This is the converse of Eercise 0) Prove that if f L, then Hint: Use the inequalit f L f L f L True or False? In Eercises 3 8, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 3 0 sin 4 5 If f g for all real numbers other than 0, and f L, 0 then 6 If f L, then f c L 7 f 3, where 8 If f < g for all a, then f < g a a 9 Think About It Find a function f to show that the converse of Eercise is not true [Hint: Find a function f such that f L but f does not eist] 0 Prove the second part of Theorem 9 b proving that cos 0 0 Let f 0,, and g 0,, Find (if possible) if is rational if is irrational if is rational if is irrational 0 g L 0 f 3, 0, f and > g 0 Graphical Reasoning Consider f sec Find the domain of f Use a graphing utilit to graph f Is the domain of f obvious from the graph? If not, eplain (c) Use the graph of f to approimate f 0 (d) Confirm the answer in part (c) analticall 3 Approimation cos Find 0 Use the result in part to derive the approimation cos for near 0 (c) Use the result in part to approimate cos0 (d) Use a calculator to approimate cos0 to four decimal places Compare the result with part (c) 4 Think About It When using a graphing utilit to generate a table to approimate sin, a student concluded that 0 the it was rather than Determine the probable cause of the error
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