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

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
Limits MA1505 Mathematics I Chapter 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
Outline 1. Definition of Limits. 2. Basic Results on Limits 3. One-sided Limits
Background image of page 2
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
Background image of page 3

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

View Full DocumentRight Arrow Icon
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
Background image of page 4
. 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
Background image of page 5

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

View Full DocumentRight Arrow Icon
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
Background image of page 6
. 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 =
Background image of page 7

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

View Full DocumentRight Arrow Icon
. 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
Background image of page 8
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
Background image of page 9

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

View Full DocumentRight Arrow Icon
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 = =
Background image of page 10
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 = =
Background image of page 11

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

View Full DocumentRight Arrow Icon
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 =
Background image of page 12
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
Background image of page 13

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

View Full DocumentRight Arrow Icon
0 Case (1). ( ) not defined. fx 0 Note : a hole at .
Background image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 48

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

This preview shows document pages 1 - 15. Sign up to view the full document.

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