Continuity - Continuity Advanced Level Pure Mathematics...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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 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 > 0 , there exists X > 0 such that when x > X , <- ) ( x f . y y l+ l+ l l- l- X x 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 x x 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 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 ) ( 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. 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 ) ( lim = x g x , then ) ( ) ( 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 ) ( 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 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. Let f(x) , g(x) , h(x) be three functions defined on R....
View Full Document

This note was uploaded on 11/26/2011 for the course COMPUTER S 1003 taught by Professor Angelosstavrou during the Spring '11 term at King Saud University.

Page1 / 27

Continuity - Continuity Advanced Level Pure Mathematics...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online