Chapter 1 Limits and Their Properties
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1 Chapter 1 Limits and Their Properties
2 Calculus: Chapter P Section P.2, P.3 Chapter P (briefly)
3 WARM-UP 1. Evaluate: cot 6 2. Find the domain of the function: f( x) 3x 3 2 x 4 g f ( x) f ( x) x 5 3. Find if and g( x) x 2 Calculus: Chapter P Section P.2, P.3
4 sin cos tan Calculus: Chapter P Section P.2, P.3
5 2 sin 3 11 csc 6 5 cot 4 7 cos 3 5 sin 6 5 tan 2 cos 7 6 Calculus: Chapter P Section P.2, P.3
6 Section P.2 Point-Slope Formula y y m x x 1 1 Perpendicular, Parallel Lines a b a m b m a m ll b Calculus: Chapter P Section P.2, P.3
7 Calculus: Chapter P Section P.2, P.3
8 Section P.3 Domain & Range of a Function RTGP!!!! Are there any no-no s? Interval Notation One-to-One Functions Horizontal Line Test Inverse Functions must be one-to-one Monotonic Calculus: Chapter P Section P.2, P.3
9 Section P.3 Composite Functions Mastery of this topic is essential for Calculus Be sure to follow the order of operations Even, Odd Functions A function is EVEN if f ( x) f ( x) A function is ODD if f ( x) f ( x) Calculus: Chapter P Section P.2, P.3
10 Find the domain and range of t yt ( ) sec 4 Calculus: Chapter P Section P.2, P.3
11 Calculus: Chapter P Section P.2, P.3
12 Calculus: Chapter P Section P.2, P.3
13 ABSOLUTE VALUE Absolute Value Functions are PIECEWISE-DEFINED functions: x x, x 0 x, x 0 Re-write the following functions as piecewise-defined functions: f ( x) 3x 6 f ( x) 4x 12 f x 2 ( ) x 1 Calculus: Chapter P Section P.2, P.3
14 ABSOLUTE VALUE EQUATIONS & INEQUALITIES To solve absolute value equations: 3x 2 7 7x 1 4 Inequalities: The Great-OR Type x a x a OR x a 2x 1 6 Calculus: Chapter P Section P.2, P.3
15 ABSOLUTE VALUE EQUATIONS & INEQUALITIES Inequalities: The Less ThAND Type x a x a AND x a We will actually represent this as follows: x a a x a x 2 1 3x 2 13 Calculus: Chapter P Section P.2, P.3
16 ABSOLUTE VALUE EQUATIONS & INEQUALITIES Representing Intervals with Absolute Value 4 x10 x 7 3 We need to develop a quick way to go from the compound inequality to the absolute value inequality: 1x 11 3 x 5 3 x 22 Calculus: Chapter P Section P.2, P.3
17 And finally What about taking the square root of something squared? 2 x x 2 ( x 9) 2 (2x 3) Calculus: Chapter P Section P.2, P.3
18 Calculus: Chapter 1 Section 1.1 SECTION 1.1 What is Calculus?
19 SECTION 1.2 Finding Limits Graphically and Numerically Calculus: Chapter 1 Section 1.2
20 f( x) x 3 1 x 1 Calculus: Chapter 1 Section 1.2
21 f( x) 1, x 1 0, x 1 Calculus: Chapter 1 Section 1.2
22 f( x) x x 11 Calculus: Chapter 1 Section 1.2
23 Calculus: Chapter 1 Section 1.2 Limits that Fail to Exist
24 Limits that Fail to Exist f( x) 1 x 2 Calculus: Chapter 1 Section 1.2
25 Limits that Fail to Exist 1 lim sin x0 x Calculus: Chapter 1 Section 1.2
26 Limits typically do not exist when: f(x) approaches a different number from the right side of c than it approaches from the left side f(x) increases or decreases without bound as x approaches c f(x) oscillates between two fixed values as x approaches c Calculus: Chapter 1 Section 1.2
27 Calculus: Chapter 1 Section 1.2
28 9 x x 8 Calculus: Chapter 1 Section 1.2
29 e-d Definition of a Limit Let f be a function on an open interval containing c (except possibly at c) and let L be a real number. The statement lim f ( x) xc Means that for each e > 0, there exists a d > 0 such that if 0 x c f ( x) L e L Calculus: Chapter 1 Section 1.2
30 Calculus: Chapter 1 Section 1.2 e-d Definition of a Limit
31 * Calculus: Chapter 1 Section 1.2
32 e-d Definition of a Limit Given f ( x) 2x 1 lim f( x) 5 x3 Find d such that f( x) and whenever x 3 Calculus: Chapter 1 Section 1.2
33 e-d Definition of a Limit Given f x lim f( x) 2 x1 2 ( ) x 1 Find d such that f( x) and whenever x 1 Calculus: Chapter 1 Section 1.2
34 Calculus: Chapter 1 Section 1.2
35 Calculus: Chapter 1 Section 1.2
36 Calculus: Chapter 1 Section 1.2
37 e-d Definition of a Limit Given lim f( x) 3 x1 f( x) Find d such that 3 x f( x) and whenever x 1 Calculus: Chapter 1 Section 1.2
38 SECTION 1.3 EVALUATING LIMITS ANALYTICALLY Calculus: Chapter 1 Section 1.3
39 Properties of Limits One of the easiest and most useful ways to evaluate a limit analytically is direct substitution If you can plug c into f(x) and generate a real number answer in the range of f(x), that generally implies that the limit exists Basic Limits: limb xc b lim xc x c lim x n c n xc Calculus: Chapter 1 Section 1.3
40 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 limits: lim f ( x) xc L lim g( x) xc K Calculus: Chapter 1 Section 1.3
41 Properties of Limits 1. Scalar Multiple: 2. Sum/Difference: 3. Product: 4. Quotient: 5. Power: lim b f ( x) b L xc lim f ( x) g( x) L K xc lim f ( x) g( x) L K xc f ( x) L lim, K 0 xc g( x) K lim ( ) n n f x L xc Calculus: Chapter 1 Section 1.3
42 Limits of Polynomials & Rationals If p is a polynomial function and c is a real number, then: lim p( x) p( c) xc If r is a rational function given by r(x) = p(x)/q(x), and c is a real number, then pc () lim r( x) r( c), q( c) 0 xc qc () Calculus: Chapter 1 Section 1.3
43 Limits of Radical Functions Let n be a positive integer. The following limit is valid for c > 0 if n is even: lim n x n c xc Calculus: Chapter 1 Section 1.3
44 Limits of Composite Functions If f and g are functions such that then, lim g( x) L and lim f ( x) f ( L) xc xl lim f ( g( x)) f lim g( x) f ( L) xc xc Calculus: Chapter 1 Section 1.3
45 Limits of Trig Functions Let c be a real number in the domain of the given trig function: 1. limsin c sin c 2. limcos c cos c xc xc 3. lim tan c tan c 4. limcot c cot xc xc c 5. limsec c sec c 6. limcsc c csc c xc xc Calculus: Chapter 1 Section 1.3
46 Strategies for Finding Limits To find limits analytically, try the following: Direct Substitution Factoring/Dividing Out Technique (S.U.A.) Rationalize Numerator/Denominator (S.U.A.) Multiplying by the Conjugate of Num/Denom. Other creative (LEGAL) versions of S.U.A. Or, you can try Calculus: Chapter 1 Section 1.3
47 x 2 x 6 lim x3 x 3 Calculus: Chapter 1 Section 1.3
48 lim x 0 x 11 x Calculus: Chapter 1 Section 1.3
49 lim x 0 x x Calculus: Chapter 1 Section 1.3
50 The Squeeze/Sandwich Theorem If h(x) < f(x) < g(x) for all x in an open interval containing c (except possibly at c), and if lim h( x) L lim g( x) xc xc lim f( x) Then exists and is equal to L. xc Calculus: Chapter 1 Section 1.3
51 Two Trig Limits (which will be easier to do in a few months ) sin x lim 1 x0 x 1 cos x lim 0 x0 x Calculus: Chapter 1 Section 1.3
52 An Application of the Squeeze Theorem cos, sin 1, tan 1, 0 Calculus: Chapter 1 Section 1.3
53 An Application of the Squeeze Theorem Area of triangle Area of sector Area of triangle Calculus: Chapter 1 Section 1.3
54 LIMIT #1 LIMIT #2 LIMIT #3 LIMIT #4 LIMIT #5 OH YEAH! OH NO! Calculus: Chapter 1 Section 1.3
55 Calculus: Chapter 1 Section 1.3
56 Calculus: Chapter 1 Section 1.4 SECTION 1.4 Continuity and One-Sided Limits
57 Left Side: Sketch the graph of a function which is continuous for all real numbers Use creativity in your sketch (push the envelope, LEGALLY) Right Side: Sketch the graph of a function which is continuous for all real numbers except at x = 3 Use creativity in your sketch (push the envelope, LEGALLY) Calculus: Chapter 1 Section 1.4
58 f(c) is not defined lim f( x) lim f ( x) f ( c) xc does not exist xc Calculus: Chapter 1 Section 1.4
59 What you are about to see is one the most important things that you will ever learn in calculus. If you forget this, I will forget you. Calculus: Chapter 1 Section 1.4
60 Definition of Continuity at a Point A function f is continuous at c if the following three conditions are met: f() c 1. is defined. lim f( x) 2. exists. xc lim f ( x) f ( c) 3.. xc DON T YOU DARE FORGET THIS. EVER. Calculus: Chapter 1 Section 1.4
61 Types of Discontinuity Removable: hole in the curve (typically) If removable, you can re-define the function to fill the hole Non-removable: gap, step, asymptote Calculus: Chapter 1 Section 1.4
62 Some Examples f( x) 1 x f( x) x 2 1 x 1 Calculus: Chapter 1 Section 1.4
63 Some Examples f ( x) sin x f( x) x x 1, x0 2 1, x0 Calculus: Chapter 1 Section 1.4
64 One-sided Limits Notation for a one-side limit typically appears as follows: Right-hand Limit: lim f ( x) xc L Left-hand Limit: lim f ( x) xc L Calculus: Chapter 1 Section 1.4
65 lim 4 x 2 x 2 lim x 2 x lim x3 x 1 3 Calculus: Chapter 1 Section 1.4
66 The Existence of a Limit Let f be a function and let c be a real number. The limit of f(x) as x approaches c is L if and only if lim f ( x ) L lim f ( x ) xc xc Remember the 5 th Grader Definition! Calculus: Chapter 1 Section 1.4
67 Definition of Continuity on Closed Interval A function f is continuous on [a, b] if it is continuous on (a, b) and lim f ( x ) f ( a ) and lim f ( x ) f ( b ) xa xb Calculus: Chapter 1 Section 1.4
68 Properties of Continuity If b is a real number and f and g are continuous at x = c, then following functions are also continuous at c: 1. Scalar Multiple: bf 2. Sum/Difference: f g 3. Product: fg f g 4. Quotient: if gc ( ) 0 Calculus: Chapter 1 Section 1.4
69 State the intervals on which f(x) is continuous f ( x) tan x f( x) 1 sin, x x 0 0, x 0 Calculus: Chapter 1 Section 1.4
70 Calculus: Chapter 1 Section 1.4
71 Intermediate Value Theorem If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that: f () c k VERY IMPORTANT THEOREM!!!!!!! Calculus: Chapter 1 Section 1.4
72 Calculus: Chapter 1 Section 1.4 f( x)
73 Use the I.V.T. to show that the polynomial function below has a zero on the interval [0, 1] 3 f ( x) x 2x 1 Calculus: Chapter 1 Section 1.4
74 3 2 f ( x) x x x 2 Explain why there must be some value c on the interval [0, 3] such that f(c) = 4. Calculus: Chapter 1 Section 1.4
75 Calculus: Chapter 1 Section AP Free-Response #3
76 Calculus: Chapter 1 Section 1.5 SECTION 1.5 Infinite Limits
77 Infinite Limits Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement: lim f( x) xc Means that for each M > 0 there exists a d > 0 such that f(x) > M whenever 0 x c *similar definition for limits approaching Calculus: Chapter 1 Section 1.5
78 Vertical Asymptotes If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f. *Be careful with rational functions that can reduce by dividing out factors Calculus: Chapter 1 Section 1.5
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