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Continuity - Continuity Advanced Level Pure Mathematics...

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Continuity Advanced Level Pure Mathematics Advanced Level Pure Mathematics 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 3.1 Introduction Prepared by K. F. Ngai(2003) Page 1 3
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1. f(x) may have a finite limit value. 2. f(x) may approach to infinity; 3. f(x) may oscillate or infinitely and limit value does not exist. Continuity Advanced Level Pure Mathematics Example f x x ( ) = 2 is continuous on R. Example f x x ( ) = 1 is discontinuous at x = 0. y y y x = 2 y x = 1 0 x 0 x 3.2 Limit of a Function A. Limit of a Function at Infinity Defintion Let f(x) be a function defined on R. = ) ( lim x f x means that for any ε > 0 , there exists X > 0 such that when x > X , ε < - ) ( x f . y y l+ ε l+ ε l l l 0 X x 0 X x N.B. (1) = ) ( lim x f x means that the difference between f(x) and A can be made arbitrarily small when x is sufficiently large. (2) = ) ( lim x f x means f(x) → A as x → ∞. (3) Infinity, ∞, is a symbol but not a real value. There are three cases for the limit of a function when x → ∞, y y y L 0 x 0 x 0 x Theorem UNIQUENESS of Limit Value If a x f x = ) ( lim and b x f x = ) ( lim , then b a = . Theorem Rules of Operations on Limits Prepared by K. F. Ngai(2003) Page 2
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Continuity Advanced Level Pure Mathematics If ) ( lim x f x and ) ( lim x g x exist , then (a) ) ( lim ) ( lim )] ( ) ( [ lim x g x f x g x f x x x ± = ± (b) ) ( lim ) ( lim ) ( ) ( lim x g x f x g x f x x x = (c) ) ( lim ) ( lim ) ( ) ( lim x g x f x g x f x x x = if 0 ) ( lim x g x . (d) For any constant k, ) ( lim )] ( [ lim x f k x kf x x = . (e) For any positive integer n, (i) n x n x x f x f )] ( lim [ )] ( [ lim = (ii) n x n x x f x f ) ( lim ) ( lim = N.B. 0 1 lim = x x Example Evaluate (a) 1 5 7 2 3 lim 2 2 - + + - x x x x x (b) ) 1 ( lim 2 4 x x x - + Theorem Let ) ( x f and ) ( x g be two functions defined on R. Suppose X is a positive real number. (a) If ) ( x f is bounded for x > X and 0 ) ( lim = x g x , then 0 ) ( ) ( lim = x g x f x (b) If ) ( x f is bounded for x > X and ± ∞ = ) ( lim x g x , then ± ∞ = ± )] ( ) ( [ lim x g x f x ; (c) If 0 ) ( lim x f x and ) ( x f is non-zero for x > X , and ± ∞ = ) ( lim x g x , then ± ∞ = ) ( ) ( lim x g x f x . Example Evaluate (a) x x x sin lim (b) x e x x cos lim - (c) ) sin ( lim x e x x ± Prepared by K. F. Ngai(2003) Page 3
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Continuity Advanced Level Pure Mathematics Example Evaluate (a) x xe x x + 1 lim (b) 1 cos lim + + x x x x Theorem SANDWICH THEOREM FOR FUNCTIONS Let f(x) , g(x) , h(x) be three functions defined on R. If a x h x f x x = = ) ( lim ) ( lim and there exists a positive real number X such that when x > X , ) ( ) ( ) ( x h x g x f , then a x g x = ) ( lim .
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