Math Analysis I - Limits of Functions Part 5 (Landau Symbols)

Local functions

of

comparison

let f , neighbourhood

defined

be a

i n

g E)

(possibly with the

I

o f exception of

ight and

l e t t o , x # I a s s u m e

fim.fi#--eeR

then said controlled

1 by

i s g

Lendeu symbols

l

If the limit is finite, we say that the function f is controlled by g for x

-

tending to x by saying that ‘f is big O of g for x tending to x ’.

-

I-0cg), X # I

This property can be made more precise by distinguishing two cases:

l

1. If is finite and non-zero, we say that f is of the same order of

magnitude as g for x tending to x :

-

I f-0cg)

f g

K X ' →

,

I

Moreover if = 1, we call the function f equivalent to g for x

tending to x :

f - I f-0cg)

g , X i u

l

2. If = 0, we say that the function f is negligible with respect to g when x

goes to x , by saying that ‘f is little o of g for x tending to x ’.

- -

0cg)

f

(g)

f - O I

↳ =

X

,

2. Math Analysis Limits of Functions 1 of 31

～

=

O o

The symbols , , , are called Landau symbols.

Little

IR, R

E IE,

E E f, E

c g: → E"

"little t o

j goes

i s o f ×

e s

g

o

symbols

i n f- 01g), E

x . →

1 ¥

is ein o

=

x - i x (x)

g 2, ( x)

e . E-o, fix) - X

g.. g-

x

= ¥-fin

EYE -Eino

¥¥, x - o

H e

then, 01 × 1, X i a o

e s

1¥.

- x 2 ,

I - t Cx)

(x) g-

f x

o o , =

• Gift-9¥, # t o

-fine, 01×4

(s), x

X # t o o

8 = 0 =

2. Math Analysis Limits of Functions 2 of 31

Proposition (reeetionshipbetweeulimitse.

ir#Leudawsymbels-)

him = L

fix) only

and if

if

E

× → L t di),

f - I

x →

Proof [ j a r l - e )

Fife ⇐ fix

11 × 1-l - o

⇐ I C H - l 011),

i X # I

= (i),

l

j r ) X # I

= t o

lying

e. size s i n e -A

g. t o l l ) ,

A

1

= X i a o

2. Math Analysis Limits of Functions 3 of 31

Proper tiesoflitt

i) (fix)),

f (x) 0111 ' →

o x o

=

I n particular

(x)

(i)

X X i - 7 0

0 o

= ,

Inde e d,

X¥ d i ) o

→

x .

→ o ,

=

g..fi#osiIx---1

e. ⇐ sine toll),

A

= X ↳ o

⇐ di),

s i n t

x x

x X

= ' → o

then, ocx),

s i n t

X o

x = X n i

X

÷i¥f÷

ex-l

lim 1

. =

I

X i a o e¥

⇐ e a t o u t X # o

,

⇐ (x)

ex 0 X - t o

I t t

x

= ,

2. Math Analysis Limits of Functions 4 of 31

l i m ' - c o s x 21

=

• 2 × 2

x → o

⇐ I - c o s I

x (i), X o

→

o

= t

X ¥

⇐ I 01×4

1 - c o s x = t

x- l - ¥ - 01 × 2 )

cos =

I - × 2 X i u

(XL), o

= t o

z

÷±#t±.

c o s I

I - I

2

ii) (i)

f - i f

E if and only

→

x .

e s

o

fizz f a r ) 0

=

¥-o

Indeed I-0181

(constant)

(x)

l e t i

g =

¥ I oh)

then =

o

=

2. Math Analysis Limits of Functions 5 of 31

iii) (rig) off) Ito,

holy) X # o

o = = ,

fi¥¥§f¥,

Indeed, suppose o

-

then, f)

o l d

g =

(x)

g (i) ( Ly n )

get

o

§ = o

=

Cx))

g a l - 1 o f f

8 ¥ = # = (i)

1

s i n c e t o , o

fat

111 × 1

o l d

off) (f)

f)

t h e n - t o

g - =

in) (nd)

(nd) (nd) 0 X # o

t

O t s

o

. . . ,

Indeed, 9

y o

61u4tug.tk#=0CuIIt... t.lu#--0111

(nd)

u'

(i).

o o

=

=

a) (XP)

x" p.

h →

s

X o o

o

= x . o

, ,

¥

Indeed, 0111 ' →

x o

= ,

01×131

⇐ x' B

i a s

= ,

2. Math Analysis Limits of Functions 6 of 31

xp),

Oi) x" o l p

X i a o a e

- x i a o

,

,

1¥

Indeed di) ' → o

x

= ,

(x-l,

⇐ x " p

d e

o

=

Vii) 01 × 41=01 × 4 p,

x → o x .

a >

,

,

Indeed,

01×4) ⇐ (XP),

old)

0111 X i a o

- O

-

x - p u p

I n particular 0¥-o,

(x)

01 × 2) since

o X

= t o

×

0¥¥

⇐ o - x o

=

-

the (x) 01×4

t r u e

opposite n o t

i s f-

0

0¥-o¥÷

+ o f r

01 × 1

be

I c a n n o t x # o

s i n c e

(x l, (x) dx4

' different

be

I from

e m a y

o

2. Math Analysis Limits of Functions 7 of 31

Viii) mink-M),

(XP) (x

0 1 × 4 o

± o →

= x . o

Indeed, p

for a >

01×41,011=4%1 ¥1...

±

01 × 4 ¥10-old x o

,

own

01 × 4 I (l)

= •

✓ XP (XP),

(XP)

01×4 P-mink. P)

± =

o o

in) 41×101×4=01×4 E - o

X - x E

if bounded neighbourhood

4 i n E

is of

a