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1、Chapter 1Limits and Their PropertiesLimits The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of ones endurance” or the “limit of ones patience.”In mathematics, the word “limit” has a similar but more precise meaning. Suppose you drive
2、200 miles, and it takes you 4 hours.Then your average speed is:If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.1.1 Rates of Change and LimitsA rock falls from a high cliff. The position of the rock is given by:After 2 seconds:average speed:Wha
3、t is the instantaneous speed at 2 seconds?1.1 Rates of Change and Limitsfor some very small change in twhere h = some very small change in tWe can use the TI-84 to evaluate this expression for smaller and smaller values of h.1.1 Rates of Change and Limits1800.165.6.0164.16.00164.016.000164.0016.0000
4、164.0002We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero.(Note that h never actually becomes zero.)1.1 Rates of Change and LimitsThe limit as h approaches zero:01.1 Rates of Change and LimitsDefinition
5、: LimitLet c and L be real numbers. The function f has limit L as x approaches c if, for any given positive number , there is a positive number such that for all x,1.1 Rates of Change and LimitsaLfDNE = Does Not ExistafL1L21.1 Rates of Change and LimitsDefinition: One Sided LimitsLeft-Hand Limit: Th
6、e limit of f as x approaches a from the left equals L is denoted Right-Hand Limit: The limit of f as x approaches a from the right equals L is denoted 1.1 Rates of Change and Limits1.1 Rates of Change and LimitsDefinition: Limitif and only if and1.1 Rates of Change and LimitsDNE = Does Not ExistPoss
7、ible Limit Situationsafaf1.1 Rates of Change and Limits123412At x = 1:left hand limitright hand limitvalue of the function does not exist because the left and right hand limits do not match!1.1 Rates of Change and LimitsAt x = 2:left hand limitright hand limitvalue of the functionbecause the left an
8、d right hand limits match.1234121.1 Rates of Change and LimitsAt x =3:left hand limitright hand limitvalue of the functionbecause the left and right hand limits match.1234121.1 Rates of Change and LimitsLimits Given a function f(x), if x approaching 3 causes the function to take values approaching (
9、or equalling) some particular number, such as 10, then we will call 10 the limit of the function and writeIn practice, the two simplest ways we can approach 3 are from the left or from the right. Limits For example, the numbers 2.9, 2.99, 2.999, . approach 3 from the left, which we denote by x3 , an
10、d the numbers 3.1, 3.01, 3.001, . approach 3 from the right, denoted by x3 +. Such limits are called one-sided limits. Use tables to findExample 1 FINDING A LIMIT BY TABLES Solution :We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from t
11、he right.20Limits IMPORTANT!This table shows what f (x) is doing as x approaches 3. Or we have the limit of the function as x approaches We write this procedure with the following notation. x22.92.992.99933.0013.013.14f (x)89.89.989.998?10.00210.0210.212 Def: We writeIf the functional value of f (x)
12、 is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c). or as x c, then f (x) L 310HLimitsAs you have just seen the good news is that As you have just seen the good news is that manymany limits can be evaluated by limits can be evaluated by direct di
13、rect substitutionsubstitution. .22Limit PropertiesThese rules, which may be proved from the definition of limit, can be summarized as follows.For functions composed of addition, subtraction, multiplication, division, powers, root, limits may be evaluated by direct substitution, provided that the res
14、ulting expression is defined.Examples FINDING LIMITS BY DIRECT SUBSTITUTIONSubstitute 4 for x.Substitute 6 for x.Examples FINDING LIMITS BY DIRECT SUBSTITUTIONExample 1 FindExample 2 Find Some algebraic rules of limits 1Example Some algebraic rules of limits 2Example Some algebraic rules of limits 3
15、Example Example 3: Find Example 4 Find if you plug in some very small values for , you will see this function approaches . And it doesnt matter whether is positive or negative , you still get , look at the graph of The denominator is positive in both cases, so the limit is the same.Example 5 Because
16、 the right-hand limit is not equal to the left-hand limit, the limit does not exist.There are some very important points that we need to emphasize from the last two examples.1) If the left-hand limit of a function is not equal to the right-hand limit of the function, then the limit does not exist.2)
17、 A limit equal to infinity is not the same as a limit that does not exist, but sometimes you will see the expression no limit, which serves both purposes. If , the limit, technically, does not exist.3)If k is a positive constant, then and does not exist. 4)If k is a positive constant, then and Examp
18、le 6: Find As gets bigger and bigger, the value of the function gets smaller and smaller. Therefore,Example 7: Find Its the same situation as the one in Example 6; as decrease (approaches negative infinity), the value of the function increase (approaches aero).We write his ,Some algebraic rules of l
19、imits 4Example 8 Find When you have variables in both the top and bottom, you cant just plug into the expression. You will get . We solve this by using the following technique:When an expression consists of a polynomials divided by another polynomials, divide each term of the numerator and the denom
20、inator by the highest power of that appears in the expression. The highest power of in this case is , so we divide every term in the expression ( both top and bottom) by , like so: Now when we talk the limit, the two terms containing approach zero. Were left with . Example 9:Find Divide ezch term by
21、 .You get: Example 10:Find Divide ezch term by . The other powers dont matter, because theyre all going to disappear. Now we have three new rules for evaluating the limit of a rational expression as approaches infinity:1)If the highest power of in a rational expression is in the numerator, then the
22、limit as approaches infinity is infinity.Example :2) If the highest power of in a rational expression is in the denominator, then the limit as approaches infinity is zero. Example :3) If the highest power of in a rational expression is the same in both the numerator and denominator, then the limit a
23、s approaches infinity is the coefficient of the highest term in the numrator divided by the coefficient of the highest term in the denomiator. Example :1.2 Limits of trigonometric functions Rule No.1 :This may seem strange, but if you look at the graphs of they have approximately the same slope near
24、 the origin(as gets closer to zero).Since and the sine of are about the same as approaches zero, their quotient will be very close to one. Furthermore, because ( review cosine values if you dont get this!), we know that Now we will find a second rule. Lets evaluate the limit First， multiply the top
25、and bottom by . We get: Now simplify the limit to : Next, we can use the trigonometric identity and rewrite the limit as : Now, break this into two limits :The first limit is -1 （see Rule No.1) and the second is 0, so the limit is 0.Rule No.2 :Example 11:Find Example 12:Find Rule No.3:Rule No.4:Exam
26、ple 13:Find Problem 1. Find Problem 2. Find Problem 3. FindProblem 4. FindProblem 5. FindProblem 6. FindProblem 7. Find Theorem 1.2 Properties of LimitsTheorem 1.3 Limits of Polynomial and Rational FunctionsUse your calculator todetermine the following:(a) (b) 1.2 Limits of trigonometric functions 1
27、DNESuppose that c is a constant and the following limits exist2.1 Rates of Change and LimitsSuppose that c is a constant and the following limits exist2.1 Rates of Change and Limitswhere n is a positive integer.where n is a positive integer.where n is a positive integer.where n is a positive integer
28、.2.1 Rates of Change and LimitsEvaluate the following limits. Justify each step using the laws of limits.16-5/4262.1 Rates of Change and Limits1.If f is a rational function or complex:a.Eliminate common factors.b.Perform long division.c.Simplify the function (if a complex fraction)2.If radicals exis
29、t, rationalize the numerator or denominator. 3.If absolute values exist, use one-sided limits and the following property.2.1 Rates of Change and Limits3/2DNE1/2DNE2.1 Rates of Change and LimitsTheoremIf f(x) g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x appro
30、aches a, then 2.1 Rates of Change and LimitsThe Squeeze (Sandwich) TheoremIf f(x) g(x) h(x) when x is near a (except possibly at a) andthen2.1 Rates of Change and LimitsShow that:The maximum value of sine is 1, soThe minimum value of sine is -1, soSo:2.1 Rates of Change and LimitsBy the sandwich the
31、orem:2.1 Rates of Change and Limits2.1 Rates of Change and LimitsTherefore,2.1 Rates of Change and Limitssimplify and divide by sin 2.1 Rates of Change and Limits2.1 Rates of Change and LimitsP(cos , sin )Q(1,0)The notationmeans that the values of f(x) can be made arbitrarily large (as large as we p
32、lease) by taking x sufficiently close to a (on either side) but not equal to a.2.2 Limits Involving InfinityafVertical Asymptote2.2 Limits Involving InfinityVertical AsymptoteThe line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:2.2 L
33、imits Involving Infinityf(x) = ln x has a vertical asymptote at x = 0 since f(x) = tan x has a vertical asymptote at x = /2 since 2.2 Limits Involving Infinity2.2 Limits Involving Infinity-x = 3x = 1Determine the equations of thevertical asymptotes of Find the limit Let f be a function defined on so
34、me interval (a, ). Then means that the value of f(x) can be made as close to L as we like by taking x sufficiently large.2.2 Limits Involving InfinityHorizontal AsymptoteLf2.2 Limits Involving Infinity2.2 Limits Involving InfinityDefinition End Behavior ModelSuppose that f is a rational function as
35、follows:Horizontal AsymptoteThe line y = L is called a horizontal asymptote of the curve y = f(x) if either or2.2 Limits Involving Infinityf(x) = e x has a horizontal asymptote at y = 0 since 2.2 Limits Involving InfinityIf n is a positive integer, then2.2 Limits Involving InfinityFind the limit2.2
36、Limits Involving Infinity-1/32/31/3Find the limit2.2 Limits Involving InfinityUse squeeze theorem2.2 Limits Involving InfinityA function is continuous at a point if the limit is the same as the value of the function.This function has discontinuities at x = 1 and x = 2.It is continuous at x = 0 and x
37、 =4, because the one-sided limits match the value of the function1234122.3 ContinuityDefinition: ContinuityA function is continuous at a number a if That is,1.f(a) is defined2. exists3.2.3 ContinuityDefinition: One Sided ContinuityA function f is continuous from the right at a number a if and f is c
38、ontinuous from the left at a if 2.3 Continuity1. Removable discontinuity2.3 Continuity2. Infinite discontinuity2.3 Continuity3. Jump discontinuity2.3 Continuity4. Oscillating discontinuity2.3 ContinuityDefinition: Continuity On An IntervalA function f is continuous on an interval if it is continuous
39、 at every number in the interval. (If f is defined on one side of an endpoint of the interval, we understand continuous at the endpoints to mean continuous from the right or continuous from the left).2.3 ContinuityTheorem1. f + g 2. f g3. cf4. fg5. f / g if g(a) 06. f(g(x)If f and g are continuous a
40、t a and c is a constant, then the following functions are also continuous at a:2.3 ContinuityTheorem(a)Any polynomial is continuous everywhere; that is, it is continuous on = (-, ).(b)Any rational function is continuous whenever it is defined; that is, it is continuous on its domain.2.3 ContinuityAn
41、y of the following types of functions are continuous at every number in their domain: Polynomials; Rational Functions, Root Functions; Trigonometric Functions; Inverse Trigonometric Functions; Exponential Functions; and Logarithmic Functions.2.3 ContinuityIf f is continuous at b and , then . In othe
42、r words, 2.3 ContinuityIf g is continuous at a and f is continuous at g(a), then the composite function f(g(x) is continuous at a.2.3 ContinuityThe Intermediate Value TheoremSuppose that f is continuous on the closed interval a, b and let N be any number between f(a) and f(b). Then there exists a nu
43、mber c in (a, b) such that f(c) = N.afbf(a)f(b)cf(c)=N2.3 ContinuityUse the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.2.3 Continuity GraphContinuous at x=0? GraphContinuous at x = 0?00yesundefined0noundefinedDNEnoundefined1no00yesundefine
44、d1noundefinedDNEno0DNEnoundefined0noDefinition: LimitLet c and L be real numbers. The function f has limit L as x approaches c if, for any given positive number , there is a positive number such that for all x,2.3 ContinuitySolution Set c = 1 and f(x) = 5x - 3 and L = 2.For any given 0, there exists
45、 a 0 such that0 |x - 1| whenever |f(x) - 2| 2.3 Continuity|(5x - 3) - 2| |5x - 5| 5|x - 1| |x - 1| 0, there exists a 0 such that0 |x - 2| whenever |f(x) - 5| 2.3 Continuity|(3x - 1) - 5| |3x - 6| 3|x - 2| |x - 2| /3So if = /32- 2 2+5+5-52.3 ContinuityDefinitionAverage Rate of ChangeThe average rate
46、of change of a quantity over a period of time is the amount of change divided by the time it takes.2.4 Rates of Change and Tangent LinesFind the average rate of change of f(x) = x2 - 2xover the interval 1,3 andthe equation of the secant line.f(1) = -1f(3) = 3(3,3)(1,-1)y = mx + b3 = 2*3 + bb = -3y =
47、 2x - 32.4 Rates of Change and Tangent Lines2.4 Rates of Change and Tangent LinesSlope of a CurveDefinition Slope of a Curve at a Point The slope of the curve y = f(x) at the point P(a, f(a) isprovided the limit existsorThe tangent line to the curve at P is the line through P with this slope.2.4 Rat
48、es of Change and Tangent LinesFind the slope of the parabola y = x2 at the point (2,4)2.4 Rates of Change and Tangent LinesDemonstration2.4 Rates of Change and Tangent LinesNormal to a curveThe normal line to a curve at a point is the line perpendicularto the tangent at that point. 2.4 Rates of Chan
49、ge and Tangent LinesFind an equation of the normal line to the curvey = 9 x2 at x = 22.4 Rates of Change and Tangent LinesAt x = 2, the slope of the tangent line is -2(2) = -4,so the slope of the normal line is .y = mx + b5= (1/4) (2) + b5= 1/2 + bb = 9/2y= (1/4) x + 9/22.4 Rates of Change and Tangent Lines
