CHAPTER 10 Limit Formulas

Transcription

1 CHAPTER 10 Limit Formulas 10.1 Definition of Limit LIMIT OF A FUNCTION (INFORMAL DEFINITION) The notation D L is read the it of f(x)asx approaches c is L and means that the functional values f(x) can be made arbitrarily close to L by choosing x sufficiently close to c. LIMIT OF A FUNCTION (FORMAL DEFINITION) The it statement D L means that for each >0, there corresponds a number υ>0 with the property that j Lj <whenever 0 < jx cj <υ A FUNCTION DIVERGES TO INFINITY (INFORMAL DEFINITION) A function f that increases or decreases without bound as x approaches c is said to diverge to infinity 1 at c. We indicate this behavior by writing (continued) 184

2 Chapter DC1 if x increases without bound and by D 1 if x decreases without bound. INFINITE LIMIT (FORMAL DEFINITION) We write DC1if, for any number N>0 (no matter how large), it is possible to find a number υ>0 such that > N whenever 0 < jx cj <υ. LIMITS INVOLVING INFINITY The it statement x!c1 D L means that for any number >0, there exists a number N 1 such that j Lj <whenever x>n 1 for x in the domain of f. Similarly x! 1 D M means that for any >0, there exists a number N 2 such that j Mj <whenever x<n 2 LIMIT OF A FUNCTION OF TWO VARIABLES (INFORMAL DEFINITION) The notation f x, y D L continued

3 186 Chapter 10 means that the functional values f(x, y) can be made arbitrarily close to L by choosing the point (x, y) close to the point x 0,y 0. LIMIT OF A FUNCTION OF TWO VARIABLES (FORMAL DEFINITION) Suppose the point P 0 x 0,y 0 has the property that every disk centered at P 0 contains at least one point in the domain of f other than P 0 itself. Then the number L is the it of f at P if, for every >0, there exists a υ>0suchthat jf x, y Lj <whenever 0 < In this case, we write f x, y D L x x 0 2 C y y 0 2 <υ 10.2 Rules of Limits BASIC RULES For any real numbers a and c, suppose the functions f and g both have its at x D c. Suppose also that both x!c1 and x! 1 exist. Limit of a constant Limit of x Scalar rule Sum rule Difference rule Linearity rule k D k for any constant k x D c [a] D a [ C g x ] D C g x [ g x ] D g x x!c1 [a C bg x ] D a x!c1 C b x!c1 g x

4 Chapter Product rules Quotient rules Power rules Limit itation theorem The squeeze rule Limits to infinity Infinite-it theorem l HOopital s rule [g x ] D [ ][ g x ] x!c1 [g x ] D [ x!c1 ][ x!c1 g x ] g x D if g x 6D 0 g x x!c1 g x D x!c1 x!c1 g x if x!c1 g x 6D 0 [ ] n [] n D n is a rational number x!c1 [] n D [ x!c1 ] n Suppose exists and ½ 0 throughout an open interval containing the number c, except possibly at c itself. Then ½ 0. If g x h x for all x in an open interval containing c (except possibly at c itself) and if g x D h x D L then D L. A x!c1 x D 0 and n x! 1 If DC1and g x D A, then [g x ] DC1and C1 if A>0 [g x ] D 1and 1 if A<0 A x n D 0 g x D g x D Let f and g be differentiable functions on an open interval containing c (except possibly at c itself). If produces an indeterminate g x form 0 0 or 1 1,then

5 188 Chapter 10 g x D f 0 x g 0 x provided that the it on the right side exists. TRIGONOMETRIC LIMITS cos x D cos c sin x D sin c tan x D tan c sec x D sec c csc x D csc c cot x D cot c sin x x!0 x D 1 x!0 sin ax x D a x!0 tan x x D 1 x!0 1 cos x x MISCELLANEOUS LIMITS ( 1 C 1 n D e n!c1 n) 1 C n!0 n 1/n D e ( 1 C k n D e n!c1 n) k (1 p C 1 ) nt D pe t n!c1 n n!c1 n1/n D Limits of a Function of Two Variables BASIC FORMULAS AND RULES FOR LIMITS OF A FUNCTION OF TWO VARIABLES Suppose f x, y and f x, y D L and Then the following rules obtain: D 0 g x, y both exist, with g x, y D M. Scalar rule Sum rule [af x, y ] D a f x, y D al [f C g] x, y [ D D L C M ] [ f x, y C ] g x, y

6 Chapter Product rule Quotient rule [fg] x, y [ D D LM [ ] f g if M 6D 0 ][ f x, y x, y D ] g x, y f x, y L D g x, y M Substitution rule If f(x, y) is a polynomial or a rational function, its may be found by substituting for x and y (excluding values that cause division by zero).