Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: swang@mun.ca Course website: http://www.ucs.mun.ca/~iangshengw/1000.html Lecture Notes for Math 1000 First Previous Net Last 1
Numerical investigation of f() = sin f() = sin f() = sin 1 0.841470985 1 0.841470985 0.1 0.998334166 0.1 0.998334166 0.01 0.999983333 0.01 0.999983333 0.001 0.999999833 0.001 0.999999833 f() = sin approaches 1 when is close (but not equal) to 0. Lecture Notes for Math 1000 First Previous Net Last 2
Graphical investigation of f() = sin 1 f() = sin approaches 1 when is close (but not equal) to 0. Lecture Notes for Math 1000 First Previous Net Last 3
Definition of limits If 1. f() is defined for all near c, but not necessarily at c itself. 2. f() L becomes arbitrarily small when is any number sufficiently close (but not equal) to c. Then we say that the limit of f() as approaches c is equal to L. We write lim f() = L. c We also say that f() L (f() approaches or converges to L) as c ( tends to c). Lecture Notes for Math 1000 First Previous Net Last 4
Limits of two simple functions For any constant k and c, we have lim k = k and lim = c. c c k c c c Lecture Notes for Math 1000 First Previous Net Last 5
Numerical investigation of lim 1 1 1 1 1 1 1 0 1 2 2.414 0.9 1.9487 1.1 2.0488 0.99 1.994987 1.01 2.004988 0.999 1.99949987 1.001 2.00049988 lim 1 1 1 = 2. Lecture Notes for Math 1000 First Previous Net Last 6
Graphical investigation of lim 1 1 1 1 2 1 lim 1 1 1 = 2. Lecture Notes for Math 1000 First Previous Net Last 7
Numerical investigation of lim 2 1 2 2 0 0 2 4 0.9 0.81 1.1 1.21 0.99 0.98 1.01 1.0201 0.999 0.998 1.001 1.002001 lim 1 2 = 1. Lecture Notes for Math 1000 First Previous Net Last 8
Graphical investigation of lim 2 1 1 1 lim 1 2 = 1. Lecture Notes for Math 1000 First Previous Net Last 9
e Numerical investigation of lim 1 0 e 1 e 1 1 1.71828 1 0.63212 0.1 1.0517 0.1 0.951626 0.01 1.005017 0.01 0.995017 0.001 1.00050017 0.001 0.99950017 lim 0 e 1 = 1. Lecture Notes for Math 1000 First Previous Net Last 10
e Graphical investigation of lim 1 0 1 1 lim 0 e 1 = 1. Lecture Notes for Math 1000 First Previous Net Last 11
Numerical investigation of lim 0 cos cos cos 1 0.5403 1 0.5403 0.1 0.995 0.1 0.995 0.01 0.99995 0.01 0.99995 0.001 0.9999995 0.001 0.9999995 lim cos = 1. 0 Lecture Notes for Math 1000 First Previous Net Last 12
Graphical investigation of lim 0 cos 1 lim cos = 1. 0 Lecture Notes for Math 1000 First Previous Net Last 13
One-sided limits Left-hand limit: left-hand side. lim f() = L if f() converges to L as approaches c from c Right-hand limit: lim f() = L if f() converges to L as approaches c c + from right-hand side. Theorem: lim f() = L if and only if both lim f() = L and lim f() = L c c c + are satisfied. Lecture Notes for Math 1000 First Previous Net Last 14
Infinite limits If f() increases without bound as c, then we write lim c f() =. If f() tends to (i.e., f() becomes negative and f() ) as c, then we write lim c f() =. If f() as approaches c from the left-hand side ( c ), then we write lim f() =. c If f() as approaches c from the right-hand side ( c + ), then we write lim f() =. c + If f() as c, then we write lim f() =. c If f() as c +, then we write lim f() =. c + Lecture Notes for Math 1000 First Previous Net Last 15
The function f() = 1 1 lim 0 =, lim 1 0 + =, lim 0 1 does NOT eist. Lecture Notes for Math 1000 First Previous Net Last 16
The function f() = 1 2 1 1 lim =, 0 2 1 lim =, 0 + 2 1 lim =. 0 2 Lecture Notes for Math 1000 First Previous Net Last 17
The function f() = ln lim ln does NOT eist, lim 0 ln =, lim 0 + ln does NOT eist. 0 Lecture Notes for Math 1000 First Previous Net Last 18
The notation The notation looks like a sideways 8: Good mathematical skills: 1 lim 8 + 8 =. 1 lim 2 + 2 = 1 lim 2 + 2 = 2 2 1 lim 2 + 2 = Lecture Notes for Math 1000 First Previous Net Last 19
An eample 1 1 1 2 1 Lecture Notes for Math 1000 First Previous Net Last 20
Basic limit laws Assume that lim f() and lim g() are finite. Then c c (i) Sum/Difference Law: lim(f() ± g()) = lim f() ± lim g(). c c c (ii) Constant Multiple Law: For any number k, (iii) Product Law: lim kf() = k lim f(). c c ( ) ( ) lim (f()g()) = lim f() lim g(). c c c (iv) Quotient Law: If lim c g() 0, then lim c f() lim f() g() = c lim g(). c Lecture Notes for Math 1000 First Previous Net Last 21
Basic limit laws The Sum and Product Laws are valid for any finite number of functions. For eample, assume that lim f 1 (), lim f 2 () and lim f 3 () are finite. Then c c c lim (f 1() + f 2 () + f 3 ()) = lim f 1 () + lim f 2 () + lim f 3 (). c c c c ( ) ( ) ( ) lim (f 1()f 2 ()f 3 ()) = lim f 1() lim f 2() lim f 3() c c c c To apply the Limit Laws, we need to assume lim c f() and lim c g() are finite. If the function at the bottom g() tends to 0, then the Quotient Law does not apply. In this case, we have to do algebraic transformation before applying the Limit Laws.. Lecture Notes for Math 1000 First Previous Net Last 22
Continuity Let f() be defined for near c AND = c. We say f is continuous at = c if lim f() = f(c). c Otherwise, we say f is discontinuous at = c. 1. We say f has a removable discontinuity at = c if lim f() f(c). c 2. We say f has a jump discontinuity at = c if lim f() lim f(). c c + 3. We say f has an infinite discontinuity at = c if either or both the one-sided limit is or. Lecture Notes for Math 1000 First Previous Net Last 23
One-sided continuity and laws of continuity f() is called left-continuous if f() is called right-continuous if lim f() = f(c). c lim f() = f(c). c + (Laws of Continuity) If f() and g() are continuous at = c. Then f() ± g(), kf(), f()g() are continuous at = c. If further, g(c) 0, then f() g() is continuous at = c. Lecture Notes for Math 1000 First Previous Net Last 24
An eample 1 1 1 2 1 f() is continuous at = 1, and discontinuous at = 1, 0, 2. f() has an infinite discontinuity at = 1, a jump discontinuity at = 0, and a removable discontinuity at = 2. f() is left-continuous (but NOT right-continuous) at = 0. Lecture Notes for Math 1000 First Previous Net Last 25
Basic functions, inverse function and composite function (Polynomial and rational functions) Let P () and Q() be polynomials. Then: 1. P () is continuous on the real line. 2. P ()/Q() is continuous at = c if Q(c) 0. (Basic functions) 1. f() = sin and f() = cos are continuous on the real line. 2. For b > 0, f() = b is continuous on the real line. 3. For b > 0 and b 1, f() = log b is continuous for > 0. 4. If n is a rational number, then f() = 1/n is continuous on its domain. (Inverse function) If f() is continuous on an interval I with range R, and if the inverse f 1 () eists, then f 1 () is continuous on R. (Composite function) If g() is continuous at = c, and if f(y) is continuous at y = g(c), then f(g()) is continuous at = c. Lecture Notes for Math 1000 First Previous Net Last 26
The squeeze theorem f f() is squeezed at = 0 by l() and u(). Lecture Notes for Math 1000 First Previous Net Last 27
The squeeze theorem f f() is NOT squeezed at = 0 by l() and u(). Lecture Notes for Math 1000 First Previous Net Last 28
The squeeze theorem If f() is squeezed at = c by l() and u(), namely, 1. l() f() u() for all close (but not equal) to c. 2. lim l() = lim u() = L. c c Then the limit lim c f() eists and lim c f() = L. The meat/vegetable is squeezed by the two breads. Lecture Notes for Math 1000 First Previous Net Last 29
The squeeze theorem: lim 0 sin = 1 u 1 l cos f() = sin is squeezed at = 0 by l() = cos and u() = 1. Lecture Notes for Math 1000 First Previous Net Last 30
The squeeze theorem: lim 0 sin 1 = 0 u f sin 1 l f() = sin 1 is squeezed at = 0 by l() = and u() =. Lecture Notes for Math 1000 First Previous Net Last 31
Three techniques for evaluating limits and two formulas Three techniques: 1. Algebraic Transformation 2. The Squeeze Theorem 3. Change of Variable Two formulas: lim 0 sin = 1 and lim 0 1 cos 2 = 1 2. Remark: the second formula implies ( 1 cos lim = lim 0 0 1 cos 2 ) ( ) lim 0 = ( ) 1 (0) = 0. 2 Lecture Notes for Math 1000 First Previous Net Last 32
Limits at infinity and asymptotes If f() L becomes arbitrarily small (f() L) as increases without bound ( ), then we write lim f() = L. If f() L becomes arbitrarily small (f() L) as decreases without bound ( ), then we write lim f() = L. The horizontal line y = L is called a horizontal asymptote of f() if lim f() = L, or lim f() = L. The vertical line = c is called a vertical asymptote of f() if either or both the one-sided limit at = c is or (namely, f() has an infinite discontinuity at = c). Lecture Notes for Math 1000 First Previous Net Last 33
Limits at infinity For any a > 0, we have lim a = 0, and lim a =. Proposition: Given positive constant L > 0, we have ± L =, + = L =, ( L) =, = L 0 + =, L 0 =, L 0 + =, L 0 = Remark: 0 0,, 0, are indeterminate. Lecture Notes for Math 1000 First Previous Net Last 34
2 runs faster than as 2 is the hare (rabbit), is the tortoise (turtle), 2 runs faster than as. lim (2 ) =. Lecture Notes for Math 1000 First Previous Net Last 35
Limits of polynomials at infinity Transform the difference/sum into a product: lim (2 ) = lim 2 (1 1 ) =. The leading term of a polynomial dominates when, namely, if n 1 and a n 0, then lim (a n n + a n 1 n 1 + + a 0 ) = lim n (a n + a n 1 1 + + a 0 n ) = lim a n n {, for a n > 0 =, for a n < 0 Lecture Notes for Math 1000 First Previous Net Last 36
Limits of rational functions at infinity and limits at Let n 1, m 1, a n 0 and b m 0, then lim a n n + a n 1 n 1 + + a 0 b m m + b m 1 m 1 + b 0 = lim = lim n (a n + a n 1 1 + + a 0 n ) m (b m + b m 1 1 + b 0 m ) a n n b m m = a n b m lim n m. If, then we make a change of variable y = (replace by y and by y): lim f() = lim f( y). y Lecture Notes for Math 1000 First Previous Net Last 37
Horizontal and vertical asymptotes The horizontal asymptotes come from the limits at infinity (if eist). The vertical asymptotes come from the points of infinite discontinuity. Find horizontal and vertical asymptotes of a rational function: 1. Evaluate the limits at and to obtain horizontal asymptotes. 2. Find the zeros of denominator and evaluate the corresponding one-sided limits to obtain vertical asymptotes (if any). Lecture Notes for Math 1000 First Previous Net Last 38