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Continuity Advanced Level Pure Mathematics Advanced Level Pure Mathematics 3 Calculus I Chapter 3 Continuity 3.1 Introduction 2 3.2 Limit of a Function 2 3.3 Properties of Limit of a Function 9 3.4 Two Important Limits 10 3.5 Left and Right Hand Limits 12 3.6 Continuous Functions 13 3.7 Properties of Continuous Functions 16 Prepared by K. F. Ngai(2003) Page 1 Continuity Advanced Level Pure Mathematics 3.1 Introduction Example f ( x ) x 2 is continuous on R. Example y f ( x) y y x2 0 3.2 1 is discontinuous at x = 0. x y x 0 1 x x Limit of a Function A. Limit of a Function at Infinity Let f(x) be a function defined on R. lim f ( x) means that for any > 0 , there exists X > 0 Defintion x such that when x > X , f (x) . y y l+ l l l+ l 0 N.B. (1) X x 0 X x lim f ( x) means that the difference between f(x) and A can be made arbitrarily small when x is x sufficiently large. (2) lim f ( x) means f(x) → A as x → ∞. x (3) Infinity, ∞, is a symbol but not a real value. There are three cases for the limit of a function when x → ∞, 1. f(x) may have a finite limit value. y 2. f(x) may approach to infinity; y 3. f(x) may oscillate or infinitely and limit value does not exist. y L 0 0 x 0 x x Prepared by K. F. Ngai(2003) Page 2 Continuity Advanced Level Pure Mathematics Theorem UNIQUENESS of Limit Value If lim f ( x) a and lim f ( x) b , then a b . x Theorem x Rules of Operations on Limits If lim f ( x) and lim g ( x) exist , then x x (a) lim [ f ( x) g ( x)] lim f ( x) lim g ( x) (b) lim f ( x) g ( x) lim f ( x) lim g ( x) (c) lim x x x x x x f ( x) f ( x) lim x x g ( x) lim g ( x) if lim g ( x) 0 . x x (d) For any constant k, lim [kf ( x)] k lim f ( x) . x x (e) For any positive integer n, (i) lim [ f ( x)]n [ lim f ( x)]n x (ii) lim x N.B. lim x x n f ( x) n lim f ( x) x 1 0 x 3x 2 2 x 7 x 5 x 2 x 1 (b) lim ( x 4 1 x 2 ) Example Evaluate (a) lim Theorem Let f (x) and g (x ) be two functions defined on R. Suppose X is a positive real number. x (a) If f (x) is bounded for x > X and lim g ( x) 0 , then lim f ( x) g ( x) 0 x x (b) If f (x) is bounded for x > X and lim g ( x) , then lim [ f ( x) g ( x)] ; x x (c) If lim f ( x) 0 and f (x) is non-zero for x > X , and lim g ( x) , then x x lim f ( x) g ( x) . x Prepared by K. F. Ngai(2003) Page 3 Continuity Advanced Level Pure Mathematics Example Evaluate (a) lim sin x x x (b) lim e x cos x Example Evaluate (a) lim xex x 1 x (b) lim Theorem SANDWICH THEOREM FOR FUNCTIONS x (c) lim (e x sin x) x x cos x x x 1 Let f(x) , g(x) , h(x) be three functions defined on R. If lim f ( x) lim h( x) a and there exists a positive real number X such that when x > X , x x f ( x) g ( x) h( x) , then lim g ( x) a . x y y = h(x) y = f(x) 0 x Prepared by K. F. Ngai(2003) Page 4 Continuity Advanced Level Pure Mathematics Example (a) Show that for x > 0, ( x 1)( x 3) ( x 1)( x 3) x 2 . x2 (b) Hence find lim [ ( x 1)( x 3) x] . x N.B. Example 1 lim (1 ) n e for any positive integer n. n n 1 Evaluate lim (1 ) x e by sandwich rule. x x Prepared by K. F. Ngai(2003) Page 5 Continuity Advanced Level Pure Mathematics 1 1 lim (1 ) x e lim (1 y) y e x y 0 x Theorem Evaluate (a) Example N.B. Exercise 1. lim (1 x) x0 1 lim (1 ) x e x x (a) lim (1 3x) x 0 x2 1 (b) lim 2 x x 1 1 x 1 x x2 1 (c) lim x 1 x x 1 1 2. lim (1 x) x e . x 0 2 (b) lim (1 ) x x x (c) lim (1 tan x) cot x x (d) lim ( x x 1 x ) x 1 Prepared by K. F. Ngai(2003) Page 6 Continuity Advanced Level Pure Mathematics B. Limit of a Function at a Point Definition Let f(x) be a function defined on R. lim f ( x) means that for any > 0 , there exists > 0 such xa that when 0 x a , f (x) . y y = f(x) f(x) l a x 0 N.B. x (1) lim f ( x) means that the difference between f(x) and A can be made arbitrarily small when x is xa sufficiently close to a. (2) If f(x) is a polynomial , then lim f ( x) f (a ) . xa (3) In general , lim f ( x) f (a ) . xa (4) f(x) may not defined at x = a even through lim f ( x ) exists. x a lim cos x cos 1 . Example lim ( x 2 2 x 5) 32 2(3) 5 8 , Example 3 if x 5 Let f (x) , then lim f ( x) 3 but f(5) = 1 , x 5 1 if x = 5 x x3 so lim f ( x) f (5) . Therefore, f is discontinuous at x 5 . x 5 Example 2 x 1, x 2 Consider the function f ( x) . x2 4, f (x ) is discontinuous at x 2 since lim f ( x) lim (2 x 1) 5 f (2). x2 x2 ( x 2)( x 2) x2 4 , then lim f ( x) lim lim ( x 2) 4 . x 2 x2 x2 x2 x2 But f(2) is not defined on R. Example Let f ( x) N.B. lim f ( x) l is equivalent to : xa (i) lim [ f ( x) ] 0 ; xa (ii) lim f ( x) 0 ; x a (iii) lim f ( x) lim f ( x); xa xa (iv) lim f (a h) . h 0 Prepared by K. F. Ngai(2003) Page 7 Continuity Advanced Level Pure Mathematics Theorem Rules of Operations on Limits Let f (x) and g (x ) be two funcitons defined on an interval containing a , possibly except a . If lim f ( x) h and lim g ( x) k , then xa x a (a) lim f ( x) g ( x) h k x a (b) lim f ( x) g ( x) hk xa (c) Example lim x a f ( x) h , (if k 0) g ( x) k Evaluate (a) (c) Example x3 1 x 1 x 1 2 x lim x2 3 x2 5 lim 1 x x 3 2 2 x 2 Evaluate (a) lim (c) lim x 2 x2 x2 2 2 (1 x) n 1 x 0 x (b) lim (d) lim ( x 2 x x) x 1 x x 1 2 2 x 2 (b) lim (d) lim x 1 1 x 1 3 x Prepared by K. F. Ngai(2003) Page 8 Continuity Advanced Level Pure Mathematics Example (a) Prove that for x ( 2 ,0) (0, 2 ) , 2 sin x tan x 2 . 1 cos x cos x sin x tan x 2. x 0 1 cos x (b) Hence, deduce that lim Prepared by K. F. Ngai(2003) Page 9 Continuity Advanced Level Pure Mathematics 3.3 Theorem Properties of Limit of a Functon Uniqueness of Limit Value If lim f ( x) h and lim f ( x) k , then h k . x a Theorem x a Heine Theorem lim f ( x) l lim f ( x n ) l where x n a as n . xa n Example Prove that lim sin Example If lim x 0 1 does not exist. x x 3 ax 2 x 4 exists, find the value of a and the limit value. x 1 x 1 Prepared by K. F. Ngai(2003) Page 10 Continuity Advanced Level Pure Mathematics Exercise 3A 1. Give an example to show that the existence of lim [ f ( x) g ( x)] doex not imply the existenceof xa lim f ( x ) or lim g ( x) . x a 2. x a Give an example to show that the existence of lim f ( x) g ( x) doex not imply the existence of lim f ( x ) or x a x a lim g ( x) . x a 3. Evaluate the following limits. x cos x 3x 2x (a) lim (b) lim x x x 1 x 1 x 1 (e) lim x 1 2 x 2 x2 (f) 2x3 x x 2 1 lim (c) x 2 3x 2 lim ( x 4 1 x 2 ) (d) lim 2 x 1 x 5 x 6 x (g) lim ( x 2 1 x) (h) lim x (1 x) 4 (1 4 x) x 0 x x4 Prepared by K. F. Ngai(2003) Page 11 Continuity Advanced Level Pure Mathematics 3.4 Two Important Limits sin x 1 x Theorem lim N.B . (1) lim Example Find the limits of the following functions: Example x 0 sin x cos x lim x x x x cos x (3) lim x 0 x tan x x 0 x (a) lim (c) lim ( x ) sec x x 2 2 x tan x lim x 0 sin x x 0 x 1 (4) lim sin x 0 x (2) lim 1 cos x x 0 x2 (b) lim (d) lim x 0 sin 4 x sin 5 x Find the limits of the following functions: (a) lim x 1 sin ( x 1) x 2 3x 2 cos ax cos bx x 0 x2 (b) lim 3 sin x sin 3x x 0 x3 (Hint: sin 3 A 3 sin A 4 sin 3 A ) (c) lim Prepared by K. F. Ngai(2003) Page 12 Continuity Advanced Level Pure Mathematics Example Let be a real number such that 0 By using the formula sin 2 k 2 sin k 1 . 2 cos , show that k 1 2 2 θ θ θ θ sin θ 2 n 1 cos cos 2 ...cos n 1 sin n-1 . 2 2 2 2 x x x Hence, find the limit value lim (cos cos 2 ... cos n ) n 2 2 2 Prepared by K. F. Ngai(2003) Page 13 Continuity Advanced Level Pure Mathematics Exercise 1. Evaluate each of the following limits, if any. sin 5 x sin 6 x (a) lim (b) lim x 0 x 0 sin 2 x 3x 2 sin x sin x (c) lim (d) lim x x x 0 x 2. Evaluate each of the following limits, if any. (a) lim (1 tan x) (c) x 2 2x lim 2 x x 1 cot x x x 2 2x 3 (b) lim 2 x x 3 x 28 x x (d) lim (cos x) cot 2 x x 0 Prepared by K. F. Ngai(2003) Page 14 Continuity Advanced Level Pure Mathematics 3.5 Left and Right Hand Limits Theorem lim f ( x) l iff lim f ( x) lim f ( x) l . Example Show that each of the following limits does not exist. x a x a (a) lim x 2 x a x2 (b) lim x 0 x x 1 1 (c) lim e x x0 1 Example By Sandwich rule, show that lim (a x b x ) x does not exist for a b 0 . Solution If a b 0 then x 0 b 1 a 1 If a b 0 and x 0 then 1 1 b If x 0 then ( ) x 1 x 1 a 1 x 1 x 1 b a (a b ) a{1 ( ) x }x a 2 x a x As lim 2 1 , by sandwich rule, x x0 1 x 1 x x0 1 1 we have lim (a x b x ) x b we have lim (a b ) x a x0 x0 1 1 b a ( )x 1 ( )x 1 a b 1 1 1 a x x x x b (a b ) b{( ) 1}x b 2 x b x As lim b 2 b , by sandwich rule, 1 Since a b, lim (a x b x ) x does not exist. x 0 ( Why ? ) Prepared by K. F. Ngai(2003) Page 15 Continuity Advanced Level Pure Mathematics Exercise 1. Determine whether the following limits. If so, find its limit. x2 x 1 (a) lim (b) lim (c) lim 1 x0 x x2 x 2 x 0 3 4x 3.6 Continuous Functions N.B. In general, lim f ( x) f (a ) . xa Definition Let f(x) be a function defined on R. f(x) is said to be continuous at x = a if and only if lim f ( x) f (a ) . xa N.B. This is equivalent to the definition : A function f(x) is continuous at x = a , if and only if (a) f(x) is well-defined at x = a , i.e. f(a) exists and f(a) is a finite value, (b) lim f ( x ) exists and lim f ( x) f (a ) . x a xa Remark Sometimes, the second condition may be written as lim f (a h) f (a ) h 0 Example 1 Show that the function f ( x) x sin x , x 0 is continuous at x = 0. 0 , x=0 Prepared by K. F. Ngai(2003) Page 16 Continuity Advanced Level Pure Mathematics A function is discontinuous at x = a iff it is not continuous at that point a. y y = f(x) There are four kinds of discontinuity: Definition f(a) (1) Removable discontinuity: o lim f ( x ) exists but not equal to f(a). x a 0 Example 2 x 1, x 2 Show that f ( x) is discontinuous at x = 2. 4 , x = 2 Solution Since f(2) = 4 and lim f ( x) lim (2 x 1) 5 f (2) , x2 x a x 2 so f(x) is discontinuous at x = 2. 1 cos x Let f ( x) x 2 , x 0 . a , x=0 Example (a) Find lim f ( x ) . x 0 (b) Find a if f(x) is continuous at x = 0. y (2) Jump discontinuity : o lim f ( x) lim f ( x) xa xa 0 Example a y = f(x) x Show that the function f(x) = [x] is discontinuous at 2. Prepared by K. F. Ngai(2003) Page 17 Continuity Advanced Level Pure Mathematics Example Find the points of discontinuity of the function g(x) = x [x]. Example x 3 2 x 1, x 0 Let f ( x) . x0 sin x, Show that f(x) is discontinuous at x = 0. y (3) Infinite discontinuity: lim f ( x) or lim f ( x) x a y = f(x) x a i.e. limit does not exist. 0 Example Show that the function f ( x) a x 1 is discontinuous at x = 0. x Prepared by K. F. Ngai(2003) Page 18 Continuity Advanced Level Pure Mathematics y f ( x) e (4) At least one of the one-side limit does not exist. 1 x lim f ( x) or lim f ( x) do not exist. x a x a 1 1 Since lim e x 0 and lim e x , Example x0 x x0 1 so the function f ( x) e x is discontinuous at x = 0. (1) A function f(x) is called a continuous function in an open interval (a, b) if it is continuous at every point in (a, b). Definition (2) A function f(x) is called a continuous function in an closed interval [a, b] if it is continuous at every point in (a, b) and lim f ( x) f (a ) and lim f ( x) f (b) . xa x b (1) (2) y y y o o a b x a b a x b f(x) is continuous f(x) is continuous f(x) is continuous on (a, b). on (a, b). on [a, b]. N.B. Example (1) f (x ) is continuous at x = a lim f ( x) f (a) lim f (a h) f (a) . (2) f (x ) is continuous at every x on R lim f ( x h) f ( x) for all x R. xa x h 0 h 0 Let f (x) be a real-value function such that (1) f ( x y ) f ( x) f ( y ) for all real numbers x and y , (2) f (x ) is continuous at x 0 and f (0) 1 . Show that f (x) is continuous at every x R. Prepared by K. F. Ngai(2003) Page 19 Continuity Advanced Level Pure Mathematics 3.7 A. Theorem Properties of Continuous Functions Continuity of Elementary Functions Rules of Operations on Continuous Functions If f(x) and g(x) are two functions continuous at x = a , then so are f ( x) g ( x) , f ( x) g ( x) and f ( x) provided g(a) 0. g ( x) 1 cos x . Since f ( x) 1 cos x and g ( x) x 2 are two functions continuous 2 x everywhere, but g(0) = 0, so h is continuous everywhere except x = 0. Example Let h( x) Example Let h( x) Theorem Let g(x) be continuous at x = a and f(x) be continuous at x = g(a), then f。g is continuous at x = a. Example sin e x , e sin x are continuous functions. Example Show that f ( x) cos x 2 is continuous at every x R. Example Let f and g be two functions defined as f ( x) 1 cos x . Since f ( x) 1 cos x and g ( x) x 2 1 ( 0) are two functions 2 x 1 continuous everywhere, so h is continuous everywhere. x if x 0 x | x | for all x and g ( x) 2 . 2 x if x 0 For what values of x is f [g(x)] continuous? Theorem Let lim f ( x) A and g(x) be an elementary function (such as sin x, cos x, sin 1 x, e x , ln x, ... x etc.) for which g(A) is defined, then lim g[ f ( x)] g[lim f ( x)] g ( A) x x Prepared by K. F. Ngai(2003) Page 20 Continuity Advanced Level Pure Mathematics Example lim cos( tan x) cos(lim tan x) Example Evaluate lim Example Evaluate lim tan ( x 1 x 1 ln( 1 x ) . x 0 x x 0 sin x ). x B. Properties of Continuous Functions (P1) If f(x) is continuous on [a, b], then f(x) is bounded on [a, b]. (P2) But f(x) is continuous on (a, b) cannot imples that f(x) is bounded on (a, b). (1) y c < f(x) < d y f(x) is not bounded on (a, b). d Example (2) c 0 a 0 b a b x x f(x) = cot x is continuous on (0, ) , but it is not bounded on (0, ). Prepared by K. F. Ngai(2003) Page 21 Continuity Advanced Level Pure Mathematics If f(x) is continuous on [a, b], then it will attain an absolute maximum and absolute minimum on [a, b]. (P3) i.e. If f(x) is continuous on [a, b], there exist x 1 [a, b ] such that f ( x 1 ) f ( x ) and x 2 [a, b ] such that f ( x 2 ) f ( x ) for all x [a , b ] . x 1 is called the absolute maximum of the function and x2 is called the absolute minimum of the function. no. abs. max. on [a, b] y o abs. max. on [c, d] | o 0 a b c d abs. min. on [c, d] abs. min. on [a, b] (P4) If f(x) is continuous on [a, b] and f(a)f(b) < 0 , then there exists c [a, b] such that f(c) = 0. y y 0 a b x 0 a b x Example Let f ( x) cos x 0.6 which is continuous on [0, 2 ] and f(0) = 0.4 and f ( 2 ) 0.6 . Hence, there exists a real number c [0, 2 ] such that f(c) = 0. Example Prove that the equation x e x 2 has at least one real root in [0, 1]. Prepared by K. F. Ngai(2003) Page 22 Continuity Advanced Level Pure Mathematics Example Let f ( x) x 3 2 x . (a) Show that f(x) is a strictly increasing function. (b) Hence, show that the equation f ( x) 0 has a unique real root. (P5) Intermediate Value Theorem If f(x) is continuous on [a, b], then for any real number m lying between f(a) and f(b) , there corresponds a number c [a, b] such that f(c) = m. y f(b) m f(a) 0 (P6) Example a c b x Let f(a) = c and f(b) = d, if f is continuous and strictly increasing on [a, b], then f continuous and strictly increasing on [c, d] ( or [d, c] ). 1 is also f ( x) x 2 is strictly increasing and continuous on [0, 5], so f 1 ( x) x is also strictly increasing and continuous on [0, 25]. Prepared by K. F. Ngai(2003) Page 23 Continuity Advanced Level Pure Mathematics Example f ( x) cos x is strictly decreasing and continuous on [0,π], so f 1 ( x) cos 1 x is also strictly decreasing and continuous on [1, 1]. Example (a) Suppose that the function satisfies f(x+y) = f(x) + f(y) for all real x and y and f(x) is continuous at x = 0. Show that f(x) is continuous at all x. (b) A function 1 sin x 1 sin x if x 0 . f ( x) x if x = 0 0 Is f(x) continuous at x = 0? If not , how can we redefine the value of f(x) at x = 0 so that it is continuous at x = 0? Prepared by K. F. Ngai(2003) Page 24 Continuity Advanced Level Pure Mathematics Exercise Example Let f(x) be a continuous function defined for x > 0 and for any x , y > 0, f(xy) = f(x) + f(y). (a) Find f(1). (b) Let a be a positive real number. Prove that f (a r ) r f (a) for any non-negative rational number. (c) It is known that for all real numbers x, there exists a sequence {xn } of rational numbers such that lim xn x . x (i) Show that f (a x ) x f (a) for all x > 0, where a is a positive real constant. (ii) Hence show that f ( x) c ln x for all x > 0, where c is a constant. Prepared by K. F. Ngai(2003) Page 25 Continuity Advanced Level Pure Mathematics Example Let f be a real-valued continuous function defined on the set R such that f(x+y) = f(x) + f(y) for all x , y R. (a) Show that (i) f(0) = 0 (ii) f(x)= f(x) for any x R. (b) Prove that f(nx) = nf(x) for all integers n. Hence show that f(r) = rf(1) for any rational number r. (c) It is known that for all x R, there exists a sequence {an } of rational numbers such that lim a n x . n Using (b), prove that there exists a constant k such that f(x) = kx for all x R. Prepared by K. F. Ngai(2003) Page 26 Continuity Advanced Level Pure Mathematics Example Let R denote the set of all real numbers and f : R R a continuous function not identically zero such that f(x+y) = f(x)f(y) for all x , y R. (a) Show that (i) f(x) 0 for any x R. (ii) f(x) > 0 for any x R. (iii) f(0) = 1 (iv) f ( x) f ( x)1 (b) Prove that for any rational number r, f (rx ) f ( x)r . Hence prove that there exists a constant a such that f ( x) a x for all x R. Prepared by K. F. Ngai(2003) Page 27 Continuity Advanced Level Pure Mathematics Example (a) Let x 1 and define a sequence a n by a n 1 1 for n 2 2a n 1 (i) Show that a n 1 and an an1 for all n . (ii) Show that lim a n 1 . 2 a1 x and a n n (b) Let f : [1, ) R be a continuous function satisfying x2 1 for all x 1. ) 2x Using (a), show that f ( x) f (1) for all x 1. f ( x) f ( (1996 Paper II) Prepared by K. F. Ngai(2003) Page 28

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