Mathematics for Computer Science
MIT 6.042J/18.062J
Asymptotic
Properties
Albert R Meyer,
April 10, 2013
Oh-props.1
The Oh’s
lemma:
If f = o(g) or f ~ g, then f = O(g)
lim = 0 OR lim = 1 IMPLIES lim 
Albert R Meyer,
April 10, 2013
Oh-props.2
The Oh’s
If f = o(g), then g O(f)
f
lim = 0
g
IMPLIES
Albert R Meyer,
g
lim
f
April 10, 2013
=
Oh-props.4
Little Oh:
Lemma:
Proof:
o(∙)
xa = o(xb) for a  b
x
1 and b - a > 0
=
b
b-a
x
x
a
so as x 
Albert R Meyer,
1
,
®
0
b-a
x
April 10, 2013
Oh-props.5
Little Oh:
o(∙)
Lemma:
ln x =
for ε > 0.
Albert R Meyer,
ε
o(x )
April 10, 2013
Oh-props.6
Little Oh:
Lemma:
0.
Proof:
1
y
ln x =
ε
o(x )
£ y for y ³ 1 so
ln z £
z
2
2
Albert R Meyer,
o(∙)
for  >
z
1
1
y
ò
dy £
ò
z
1
y dy
for z ³ 1
April 10, 2013
Oh-props.7
Little Oh:
Lemma:
0.
Proof:
ln x =
ε
o(x )
o(∙)
for  >
, so let z =
d ln x
x
d
x
£
2
2
d
x
e
ln x £
= o(x ) for  > .
d
d
Albert R Meyer,
April 10, 2013
Oh-props.8
Little Oh:
o(∙)
Lemma:
c
x
=
for a > 1.
Albert R Meyer,
x
o(a )
April 10, 2013
Oh-props.9
Little Oh:
o(∙)
proofs:
L’Hopital’s Rule,
McLaurin Series
(see a Calculus text)
Albert R Meyer,
April 10, 2013
Oh-props.10
Big Oh:
O(∙)
Equivalent definition:
f(n) = O(g(n))
c,n0 n
n 0.
f(n) c·g(n)
Albert R Meyer,
April 10, 2013
Oh-props.11
Big Oh:
O(∙)
f(x) = O(g(x))
↑
log
scale
↓
green stays
below purple
from here on
c· g(x)
ln c
f(x)
no
Albert R Meyer,
April 10, 2013
Oh-props.12
Why limsup?
If f  2g then f = O(g),
but maybe f/g has no limit.
æ
æ
æ
æ
example
2 np
f(n) = æ1 + sin æ ææ æg(n)
æ 2 ææ
æ
but
f(n)
limsup
n®¥ g(n)
Albert R Meyer,
=2<¥
April 10, 2013
Oh-props.13