Math Analysis I - Limits of Functions Part 3 (Limit of Functions)

Limits functions

o f to}

IR.

E c EE

let

the defined behaviour

LIMIT t h e

is of

a s

f approaches "arbitrarily close"

se

e s 112

I

point

a in

FILE-IN, 7

I - z

it not t o arbitrarily

i s close

possible go

E

t o I being

without

remaining i n , it

exactly E, t h a t

s u c h i s

meaningless attempt t o compute

a n y

l i m i t 2

x .

t h e a s →

÷ . t o

→ 1

1

both and close

1 n o t arbitrarily

3 a r e

( (4)

they distance

sixed

e t

s i n c e e

a r e

2. Math Analysis Limits of Functions 1 of 41

fat-¥2,41

e . g. defined

the domain i s

natural a s

IR, the

for

where, e x p re s s i o n

X E

¥2,41 defined

is +21×2-a)

domf ± o

= [2,

{o}

-2]

f o , too)

E : -X U

U

E

± : #

It it

t o

possible t o

i s approach

distance?

arbitrarily

a n short

I

if I - - 2

= 3 yes

o r

I - o n o t

if it possible

is

no, s i n c e 0

approach

t o without

being e x a c t l y 0, E E

I

Given E # R ,

E . L i m

f : fixt

I

x →

i) N o t for domf

I

all it t o

possible

E i s

lointiousexemplee)

I

approach

2. Math Analysis Limits of Functions 2 of 41

if Not all domf

points i n

epproecuebee a r e

fam-log 1×4×2-n))

e . g. -2)

d o m f - f - o o , too)

(2,

u

: # Iz-2

I, = - 2 ,

Idea t o

possible

though is

i t

even n o t I, from

approach I,

neither n o r

both consider

sides, possible

it t o

is

interval around E i n

a n w h i c h

t h e r e will E

always b e point i n

a

( s h o r te r

a t shorter)

arbitrary a n d

a n f ro m I

distance E t o

I - O

¥f¥¥¥i¥

E E

X

2. Math Analysis Limits of Functions 3 of 41

Accumulation point

let R, AC C U M U L AT I O N

E POINT

E said

is e u

C

Of E it I x

to> Ix-Elt

E E or

t h a t

such o r

o ,

Ix-Elio x # I

Ix-El close

arbitrarily

t o I

t o

x

Definition denotes 2E

Given by the set

E , one

Pi / E}

{ x e of

point

a c c u m u l a t i o n

i s

x a n

I t said DERIVED

is SET,

t h a t

s u c h

the accumulation

a l e

of

s e t points.

Rental

III d e n o te s REAL LINE

EXTENDED

the

IT ER.

too},

{ - n ,

R tx - n e x t t o

t h a t U

such =

E

if bounded,

n o t

is upper

• then E

a c c u m u l a t i o n

i s f o r

t a r point

e u lower

E

if bounded,

n o t

i s

• t h e n - o r E

a c c u m u l a t i o n

i s f o r

point

e u

2. Math Analysis Limits of Functions 4 of 41

Limit functions

convergent

of IR,

E

let t E ,

point

accumulation o f

c o u

E # R

T: limflx) R

then, - l if

E

I

× →

te> Fo> t h e t

such

o , o 111×1-LITE

Ix-Elt

and o

E t

O

X E ACCUMULATION POINT

let

'¥÷i÷÷÷#¥

E

÷÷.

The limit does not consider the behaviour of the function f at x ,

it is not interesting.

Namely, the behaviour of the function at the precise point x

plays no role in the notation of limit.

2. Math Analysis Limits of Functions 5 of 41

Prove t h a t :

e . g . (E-1121

R

t h e

E-E',

fine

1 by l i m i t

d e f i n i t i o n of

bet 8 7 0

E y finding

w e a t

a i m

o , oilx-Iler

t h a t

such 21

1 × 2 - I

then, e E

⇐ x - E l s e

txt Ill

Consider d e l X-El-

2 1 - I x

1 × 2 - I Ell

then, t

Ix IIIx-El-

I - I t

+

= 2511

I x - I t X - I l e

= ( I x - E l t H E I K X- E )

e t

i@t 21×-1/1 x -E) I

21×-1)

(It

Ix-El

I E E

I,¥* e d i t i o n )

1 × 2-Et

if 0 - E

, tf,)

( 1,

toke 8 : - m i n

2. Math Analysis Limits of Functions 6 of 41

R

f¥¥

2- I

s i n s i n x e

x =

by definition l i m i t

of

l e t E > 0 s i n ' ¥ × ¥

21

El

Is It

i n x - s i n = c o s

21 XIII

III. los

= s i n I

s i n c e t ,

s

• h

2IsinEl.1osE1I2lsinXzIl

siualsindH4Y@H.i.÷÷

÷÷: sin'¥l 211k¥

21 l - I x - I k e

I

take O - E Ix-El-O-E

I l e

lsiux-Siu

2. Math Analysis Limits of Functions 7 of 41

Limitofconvezgentfunctionsfortt

112

E bounded

n o t upper

c pi t o }

h

E i

c He

R

l

then, Lim fan is s o

E

=

X t o Ital-else

F I I

E t h a t x >

such

e

Limitofconvezgentfunctionsfortt

112

E lower bounded

n o t

c pi l - o }

E i

c HE

R

l

then, Lim fan is

E

= s o

x - a Ifk) - f l e e

F I I

E t h a t x e

such

c-

I t possible

not account the

t a k e into

t o

i s Ix-Ile es since E - t a

o r x

expression a INDETERMINATE

)

(oo-• FORM

2. Math Analysis Limits of Functions 8 of 41

Limit divergent f u n c t i o n s

of

Positivelydivergent

R,

E R

E f :

IE, E

C E →

him fix) if

t o o

=

I

× →

FM Foto

R t h a t

such

E far)>

Ix-Elt M

E, o

o r

× e

Foresta

pi ( n o t

{ to }

E bounded 1

u p p e r

c i Lim

then, t h e R

i f

felt t o o

X t o o

→

F i re I

E that Te x t >

X > M

s u c h

F o r t

pi fo} (not bounded)

E lower

c i Lim

then, t h e R

i t

felt t o

N

-

FREE I

that Te x t >

X i M

s u c h

2. Math Analysis Limits of Functions 9 of 41

Negetivelydivergent

IR, R

E IE, f:

E E E

c →

Liff I N if

= - a s

FM IR, Fo t h e t

E such

i o M

Ix-El far)

E, t o =

X E o r

F