m M
X ‚" R
(i) x e" m; (ii) x d" M.
x"X x"X
X
m d" x d" M.
m,M"R x"X
a X ‚" R
x e" a i x0 < a + µ.
x"X µ>0 x0"X
b X ‚" R
x d" b i x0 > b - µ.
x"X µ>0 x0"X
X inf X sup X inf X =
-" sup X = " X
n
n an bn (an)
{an}
(-1)n
an =
2n
1 1 1 1
- , , -1 , , -
2 4 6 8 10
b1 = 1, b2 = 1, bn+2 = bn+1 + bn
1, 1, 2, 3, 5, 8, 13, 21, ...
(cn)
2, 3, 5, 7, 11, 13, 17, ...
" (an) {an}
" (an)
an < an+1 an d" an+1.
n"N n"N
" (an)
an+1-an (bn)
bn+1
bn
"
an
bn
cn
an a " R
lim an = a (lub lim an = a)
n"
|an - a| < µ.
µ>0 n>n0
n0"N
(-1)n
lim = 0
2n
dn = (-1)n n
n+1
an
" lim an = "
an > M
M>0 n0"N n>n0
 -" lim an = -"
an < -M
M>0 n0"N n>n0
lim(n - n2) = -"
an = (-2)n
(an) (bn)
lim(an Ä… bn) = lim an Ä… lim bn,
lim(c · an) = c · lim an c " R
lim(an · bn) = lim an · lim bn
an lim an
lim = lim bn = 0

bn lim bn
lim(an)k = (lim an)k k " N
"
"
k
k
lim an = lim an
3n3 - n2
an = bn = ( n2 + 10 -
n3 + 5n2 + n
4n + 2n
n) cn =
22n + 3n
1
1
liman = Ä…" lim = 0 ( = 0)
an Ä…"
1
lim an = 0 an > 0 (an < 0) lim = " (-") (01 = "
+
an
1
= -")
0-
a + " = " a · " = " "a = " a > 0 "a = 0
a < 0 a" = " a e" 1 a" = 0 a " (0, 1)
lim an = lim cn = b an d" bn d"
cn n " N lim bn = b
"
n
lim n = 1
"
n
lim a = 1 a > 0
"
n
an = 2n + 5n
 lim an = " an d" bn n " N
lim bn = "
1
e xn = (1 + )n
n
1
e := lim(1 + )n.
n
e H" 2, 72 H" 2, 718281828459045...
n
1 1 n+2
lim(1 + )n lim(1 - )n lim
2n n n+1
(an) (nk)
(an ) (an)
k
(an) a
a
lim an = 0 lim bn = Ä…"
1 1
n
an
lim(1 + an) = lim(1 + )b = e.
bn