! 
! 
2
{an}
α
ε > 0, n0 N,
n
N, (n ≥ n0 → |an− α| < ε )
limn→∞an = α
{an}
M > 0,
n0
N,
n
limn→∞an = +∞
N, (n ≥ n0 → M < an)
A R
A R
A R
A R x A, x ≤ a
a
A R
x A, a ≤ x
A
A
a
A
A
A R
A R
A
A
A
A A
! 
! 
! 
{an} A3
…
Am
…
supA1 ≥ supA2 ≥ supA3 ≥ … ≥ supAm ≥ … (
)
infA1 ≤ infA2 ≤ infA3 ≤ … ≤ infAm ≤ …
)
{an} {an}
{an} {an}
A2
Am = {am, am+1, … } (m=1,2,…)
(
{an} limsupn→∞an = imm→∞sup Am
{an}
liminfn→∞an = imm→∞inf Am
inf A
supA = +∞
inf A = −∞
A1
sup A
! 
{an}
limsupn→∞an = +∞
liminfn→∞an = −∞
±∞
{an}
Og
! 
! 
,
O(g)
') ∃c > 0, ∃n ∈ N, ∀n ∈ N,
0
O(g) = ( f
)* n ≥ n0 → f (n) ≤ cg(n)
f: N → [0, +∞)
n
n ≥n0
c
+)
,
)-
f(n)≤cg(n)
n0
f ∈O(g)
f(n) = O(g(n))
f≤g
10
og
3n2
,
') ∀c > 0, ∃n ∈ N, ∀n ∈ N,
0
o(g) = ( f
)* n ≥ n0 → f (n) ≤ cg(n)
T(n)
c
n
n
5
n ≥5
T(n)≤3n2
o(g)
f ∈o(g)
f(n) = o(g(n))
n ≥n0
n0
+)
,
)-
f(n)≤cg(n)
f<g
12
of,g
n
f (n ) > 0, g (n ) > 0
f ∈ o(g ) ⇔ lim
n →∞
1 2
n − 3n = O(n 2 )
2
,
f ( n)
=0
g ( n)
n
n0
n
n0
1 2
n − 3n ≤ cn 2
2
1 3
− ≤c
2 n
n ≥1
1
c = , n0 = 1
2
1 2
n − 3n = O(n 2 )
2
1 2
n − 3n ≤ cn 2
2
n ≥ n0
c
c≥
n
1
2
c
6n3 ≠ O(n 2 )
( )
6n 3 = O n 2
1 2
n − 3n ≤ cn 2
2
n ≥ n0
c
1-1 (big-O
! 
6n3 = O(n2)
O
n
n0
n≥n0
n
6n3 ≠ O(n2)
! 
)
6n3 ≤ cn2
c
6n ≤ c
a. 
2n=O(n)
b. 
n2=O(n)
c. 
n2=O(n log2n)
d. 
n log n=O(n2)
e. 
3n=2O(n)
f. 
22 =O(22 )
n
n
2O(f)
g
1-2 (small-o
h
O(f),
n
)
f, g: R→ R
a. 
n=o(2n)
b. 
2n=o(n2)
c. 
2n=o(3n)
d. 
1=o(n)
e. 
n=o(log n)
f. 
1=o(1/n)
N, g(n)=2h(n)
(1) limx→∞f(x) = limx→∞g(x) = 0
(2) x→∞
(3)
(1)(2)(3)
±∞.
f’(x)/g’(x)
x R
g’(x) ≠ 0.
limx→∞f(x)/g(x) = limx→∞f’(x)/g’(x)

1 放物型偏微分方程式における特異性の解析 レポート課題 1．G(x, y, t