# Chapter 1(A) - Limits MA1505 Mathematics I Chapter 1...

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Limits MA1505 Mathematics I Chapter 1

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Outline 1. Definition of Limits. 2. Basic Results on Limits 3. One-sided Limits
Limits In the concept of limits , we are interested in the behaviour of the values of when get closer and closer to some number . ) ( x f x a When we consider limits, the value of when is not important. In fact, need not be defined. ) ( x f ) ( a f a x = Important Remark

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Example 1. . sin ) ( Let x x x f = . 0 0 0 0 sin ) 0 ( : Note = = f defined. not is ) 0 ( So f . } 0 : { of Domain = x x f R I 0. to close is when ) ( of values the of behaviour at the look now shall We x x f
. sin ) ( Let x x x f = . } 0 : { of Domain = x x f R I 999999998 0 0001 0 999999833 0 001 0 999983333 0 01 0 sin . . . . . . x x x 1. approaches sin of value the , 0 closer to and closer get that when see we So x x x 999999998 0 0001 0 999999833 0 001 0 999983333 0 01 0 sin . . . . . . x x x - - - 0 x x 0 x < 0 x

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1. approaches sin of value the side, either from 0 closer to and closer get when graph that the from see also can we , sin ) ( of graph plot the we If x x x x x x f = . sin ) ( x x x f = 0
. 0 closer to and closer get when 1 approaches sin ) ( of value that the know we So x x x x f = ' .' 1 to equal is 0 to tends as ) ( of limit the ' ' x x f say that we case, In this : notation following the use also We . 1 ) ( lim 0 = x f x undefined. is ) 0 ( but 1 ) ( lim case, in this : Note 0 f x f x =

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. 2 to tends as ) ( of behaviour the Describe x x f Example 2. 2 if 2 if 1 3 ) ( Let = + = x x x x f y x 5 2 0 1 5. approaches ) ( of value the ), 2 to equal not (but 2 closer to and closer gets that when see we , ) ( of graph the From x f x x f : have e notation w In . 5 ) ( lim 2 = x f x ). 2 ( ) ( lim Also limit. on the effect an have not does 1 ) 2 ( of value the : Note 2 f x f f x = 3 + = x y
Example 3. . 1 ) ( Let x x f = . } 0 : { of Domain = x x f R I y x 0 value. any o approach t not does ) ( of value the , 0 closer to and closer gets that when see we , 1 ) ( of graph the From x f x x x f = exist. not does ) ( lim say that we case, In this 0 x f x

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Remarks exist. not may ) ( lim 1. x f a x ). ( number by the affected not is it exists, limit the If 2. a f ). ( to equal be not need ) ( lim 3. a f x f a x ). 2 ( ) ( lim so , 1 ) 2 ( but 5 ) ( lim have we 2, Example In 2 2 f x f f x f x x = =
Remarks exist. not may ) ( lim 1. x f a x ). ( number by the affected not is it exists, limit the If 2. a f defined. be not need ) ( 4. a f ). ( to equal be not need ) ( lim 3. a f x f a x defined. not is ) 0 ( but 1 ) ( lim sin ) ( have we 1, Example In 0 f x f x x x f x = =

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Limits (An Informal Definition). ' .' is approaches as ) ( of limit the ' ' read is which , ) ( lim write we and number the is to tends as ) ( of limit the say that then we , to close ly sufficient is when to close arbitrary gets ) ( If itself. at possibly except , containing interval open an on defined be ) ( Let 0 0 0 0 0 0 L x x x f L x f L x x x f x x L x f x x x f x x =
0 lim ( ) xx f xL = 0 What happen at is not important. = 0 ( ) need not be defined. fx L 0 x () y fx = y x

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0 Case (1). ( ) not defined. fx 0 Note : a hole at .
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## This note was uploaded on 11/01/2011 for the course MATH 1505 taught by Professor Yap during the Spring '11 term at National University of Singapore.

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Chapter 1(A) - Limits MA1505 Mathematics I Chapter 1...

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