2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril close to L (as close as we like) b taking to be sufficientl close to c (on either side of c) but possibl not equal to c. Eample 1. Evaluate the function e 1 at several points near = 0 and use the result to 2 estimate e 1 lim 0 2 Methods for Finding Limits To calculate limits we ma use 1. a table of values. 2. a graph. 3. algebra. 1
2 2 Limits and Continuous Functions Notation. The statement, the limit of f() as approaches c from the left is L is written lim f() = L. Similarl, the limit of f() as approaches c from the right is L is written f() = L. We call these statements the one-sided limits or the left and right hand + lim limits respectivel. Eample 2. Show that lim 0 does not eist. 1 Eample 3. Discuss the eistence of lim 0 2.
3 2.2 Introduction to Limits Eample 4. Discuss the eistence of lim 0 sin π. Noneistence of a Limit When a limit fails to eists, then the function ma have one of the following problems: 1. f() approaches a different value from the right side of c than it approaches from the left side. 2. f() increases or decreases without bound as approaches c. 3. f() oscillates between two fied values as approaches c.
4 2 Limits and Continuous Functions Definition. Let f be a function defined on an open interval containing c (ecept possibl at c) and let L be a real number. We sa that the limit of f() as approaches c is L and write lim f() = L provided that for each ε > 0 there eists a δ > 0 such that if 0 < c < δ, then f() L < ε.
5 2.2 Introduction to Limits Eample 5. Given lim 3 (2 1) = 5, find δ such that (2 1) 5 < 0.01 whenever 0 < 3 < δ. Eample 6. Prove that lim 3 (2 1) = 5.
6 2 Limits and Continuous Functions Eample 7. Prove that lim 3 2 = 9.
7 2.3 Limit Theorems 2.3 Limit Theorems Theorem 2.1. Let k and c be real numbers and let n be a positive integer. 1. lim k = 2. lim = 3. lim n =
8 2 Limits and Continuous Functions Theorem 2.2. Let k and c be real numbers and let n be a positive integer. Suppose that f and g are functions such that lim f() = L and lim g() = M 1. lim [ kf() ] = 2. lim [ f() ± g() ] = [ ] 3. lim f()g() = [ ] f() 4. lim = g() 5. lim [ f() ] n = Eample 1. Evaluate using the properties of limits. lim 1 (2 5 3 + 4) 3 + 2 2 1 Eample 2. Evaluate using the properties of limits. lim 2 5 3
9 2.3 Limit Theorems Theorem 2.3. 1. If p is a polnomial function and c is a real number, then lim p() = 2. If r is a rational function given b r() = p()/q(), where p and q are polnomials, and c is a real number such that q(c) 0, then lim r() = Eample 3. Evaluate the following limit. lim 1 2 + 2 + 1 + 1 Theorem 2.4. Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even: lim n = Theorem 2.5. Suppose that f and g are functions such that lim g() = M and lim f() = L M lim f( g() ) = Eample 4. Evaluate the following limit. lim 1 2 + 9
10 2 Limits and Continuous Functions Theorem 2.6. Let c be a real number in the domain of each of the following functions: 1. lim sin = 3. lim tan = 5. lim sec = 7. lim a = 2. lim cos = 4. lim cot = 6. lim csc = 8. lim ln = Eample 5. Evaluate. lim e (5 + ln ) Eample 6. Evaluate. lim 1 ln cos Theorem 2.7. Let c be a real number and let f() = g() for all c in an open interval containing c. If lim g() eists, then also eists and lim f() = lim f() Eample 7. Evaluate. lim 3 2 6 3
11 2.3 Limit Theorems Eample 8. Evaluate. lim 0 2 + 9 3 2 Theorem 2.8 (The Squeeze Theorem). If g() f() h() for all in an open interval containing c, ecept possibl at c itself, and if lim g() = eists, then lim f() =
12 2 Limits and Continuous Functions Theorem 2.9. 1. lim sin 1 cos = 2. lim =
13 2.3 Limit Theorems Eample 9. Evaluate. lim 0 tan Eample 10. Evaluate. lim 0 sin(5)
14 2 Limits and Continuous Functions 2.4 Continuit Definition. A function f is said to be continuous at a point c if the following conditions are met: 1. f(c) 2. lim f() 3. lim f() Furthermore, a function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that is continuous on (, ) is said to be continuous everwhere. Note. Another wa to understand continuit is to consider where a function is discontinuous. This is done b looking at the negations of the above definition.
2.4 Continuit Note. Discontinuities fall into two categories: 1. Removable. Algebraicall: Graphicall: 2. Nonremovable. Algebraicall: Graphicall: Eample 1. Discuss the continuit of f() = 1 1 Eample 2. Discuss the continuit of g() = 2 4 2 Eample 3. Discuss the continuit of h() = { 1 1 log > 1 Eample 4. Discuss the continuit of = cos 15
16 2 Limits and Continuous Functions Recall the use of one sided limits. Eample 5. Evaluate. lim 3 + 9 2 Definition. The greatest integer function, denoted b [[]], is the greatest integer n such that n.
17 2.4 Continuit Theorem 2.10. Let f be a function and let c and L be real numbers. Then lim f() = L if and onl if lim f() = L and lim f() = L + Definition. A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and f() = f(a) and lim f() = f(b) + lim a The function f is continuous from the right at a and continuous from the left at b. b Eample 6. Discuss the continuit of f() = 9 2. Theorem 2.11. Let k be a real number and let f and g be continuous at = c. The following functions are also continuous at c. 1. 2. 3. 4.
18 2 Limits and Continuous Functions Theorem 2.12. If g is continuous at and f is continuous at g(c), then, ( f g ) () = f ( g() ) is continuous at c. Eample 7. Discuss the continuit of f() = cot. Eample 8. Discuss the continuit of g() = { cos 1 0 0 = 0. Eample 9. Discuss the continuit of h() = { cos 1 0 0 = 0.
19 2.4 Continuit Theorem 2.13 (The Intermediate Value Theorem). If f is continuous on [a, b] and k is an number between f(a) and f(b) (in other words f(a) k f(b) or f(b) k f(a)), then there eists at least one number c in [a, b] such that f(c) = k. Note. The theorem implies that on an great circle around the world, the temperature, pressure, elevation, carbon dioide concentration, or anthing else that varies continuousl, there will alwas eist two antipodal points that share the same value for that variable. Eample 10. Use the Intermediate Value Theorem to show that the polnomial function has a zero in [0, 3].
20 2 Limits and Continuous Functions 2.5 Infinite Limits Eample 1. Construct a table of values to determine the behavior of f() = 1 1 1. as approaches Definition. Let f be a function that is defined at ever real number in some open interval containing c (ecept possibl at c itself). The statement lim f() = means that for each M > 0, there eists a δ > 0 such that f() > M whenever 0 < c < δ. Definition. If f() approaches infinit (or negative infinit) as approaches c from the right or the left, then the line = c is a vertical asmptote of the graph of f. Theorem 2.14. Let f and g be continuous on an open interval containing c. If f(c) 0, g(c) = 0, and there eists an open interval containing c such that g() 0 for all c in the interval, then the graph of the function given b h() = f() g() has a vertical asmptote at = c. Eample 2. Find all vertical asmptotes of f() = 1 2
21 2.5 Infinite Limits Eample 3. Find all vertical asmptotes of g() = 2 + 4 2 4 Eample 4. Find all vertical asmptotes of h() = 2 + 2 3 2 1 Eample 5. Find all vertical asmptotes of = tan
22 2 Limits and Continuous Functions Theorem 2.15. Let c and L be real numbers and let f and g be functions such that Then 1. lim [ f() ± g() ] = 2. lim [ f()g() ] = 3. lim g() f() = lim Note. The third item gives a special asmptote. f() = and lim g() = L. Eample 6. Suppose that f() = 3, g() = 1 4, and h() = 1. We illustrate item two as follows: