MathematicallyThe Precise Definition of a Limit vague

im thx L Hx approaches L as x approaches butc a does not equal a From both sides

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a Plot all points x y such that ly 21 a O l

b Plot all points x y 1 such that I x 11 a o Z

ly z l a o I 2 o 1 CyC 2 0.1

I 9 2 I

Ix l C 0.2 I o z x c l to 211 at

O S l Z

y

2 i ly 21 C O l

z L d strip centered Z

o g

tI l Zla l C 0.2

Key ObservationVertical strip centered at 1

Given anyhorizoutalstripecentered at C the graph

Lim 1 x Ly fGc is completely contained in

a

the strip it a is close enoughbut not equal to a

Example y fcx contained in horizontal strips

e

Tax

Hence given anghorizontalstripe centered at L

there exists a vertical strip centered at a

Such that within the vertical strip the graph y tCxRta

iscomp awd.mu setiou

Eixample

y

Ht

d x

Important Many possiblevertical strips will work

just lower the width What matters

is that at least one exists

Non example Lim 764 f Cx sa

A way outsideintersection

For

yany

choice of vertical strip

HT

d x

et's g 7 this carefully

y

E

it

d xa S a S

Epsilon For error Delta for difference

e eE o S o

77C x in horizontal x 7 x in vertical

strip stripL E tea Lt E a S C x at S

Item LICE Ix al a s

More Precise Definition of Lim 7 Cx Lx a

Given anyhorizontal strip Lcentered at c there

exists a vertical strip centered at a Sadr twat

K tea in vertical strip x 7 Cx in horizontal stripand a f a

Picture

Ct E

L

Htt

I2C a

Most Precise Definition of ingatcx L

a free choice A horizontal strip a vertical stripd

Given E o there exists 8 o which depends on E

Such that

O Ix al as 17cal Ll L E

T Tx 7 x in vertical strip Cx fc in horizontal

Cta strip

Important E o is a completely Free choice To be

Sure Lim 7 Cx L we need to Find a different S o

K aT

For each E o could be reallychallenging

Example Lim 2x 2x I

Fix E o Try 8 Ez o

y y 2x Need to be sure

2 E

µOC Ix 11CEz 12 21 as

z E ButN 0 1 11 c Ez 4x il CEi E l it E

z z 12 Ge 1 ICEtzu el CELim 2x 21

Remarks

Tis example was relatively straightforward as g tcx

was a straight line IF not it could be much

more challenging Eg Ling 5 1

7 Every limit can property can be rigorously demonstrated

using this E 8 Language

31 If we replace 0C Ix ales with a Cocca 8

we get Lim Hx Lx at

If we replace O Ix al c S with a s ca ca

we get Lim 7 x Lx a

Precise Definition of Lim 7 Cx a2

Givenany

1470 there exists 870 which depends on M

such thatfore x 7 x

0 a la al a s FCK M in vertical stripX ta 7 x M

Htt

s

Important M 0 is as big as we want hence

Fix grows positively without bound

Reinardwe define Lim 7 x a replacing M o with NCO7C a

and thx M with 7 x N