"
"
"
"
 n
n an bn
(an) (bn)
(n, an) n " N
an = 2n
1
bn =
sin n
cn = 1 + 2 + 3 + · · · + n
a1 = 7, an+1 = an + 3
b1 = 1, bn+1 = 2bn
c1 = 2, cn+1 = c1c2 · · · · · cn
an n
Ä„
bn n
(an) a " R
lim an = a,
n"
"µ>0"n "n>n |an - a| < µ.
0 0
a
a
(an) "
lim an = ",
n"
"µ>0"n "n>n an > µ.
0 0
(an) -"
lim an = -",
n"
"µ>0"n "n>n an < µ.
0 0
"
-"
an = (-1)n
nĄ
bn = sin " -"
2
" -"
a
lim = 0 a " R
n"
n
n
lim = " a " R
n"
a
"
lim n = 1
n"
"
n
lim n = 1
n"
lim pn = 0 p " R |p| < 1
n"
(an) (bn)
lim (an + bn) = lim an + lim bn
n" n" n"
lim (an - bn) = lim an - lim bn
n" n" n"
 lim (c · an) = c · lim an c " R
n"
n"
lim (an · bn) = lim an · lim bn
n" n" n"
lim
an n" an
lim =
n"
bn lim bn
n"
p
lim (an)p = lim an p " Z\ {0}
n"
n"
"
k
k
lim an = lim an p " N\ {1}
n" n"
n2 - 3n3
lim
n"
n
"3 + 1
lim n2 + n - n
n"
1 + 2 + · · · + n
lim "
n"
9n4 +
51
2n - 1
lim
n"
3n + 2
"
3
(n + 1) 8n3 + 1
lim "
n"
n n2 + 1
(n + 1)! - n!
lim
n"
(n + 1)! + n!
(an), (bn), (cn)
an bn cn n n0
lim an = lim cn = b
n" n"
lim bn = b.
n"
"
n
lim 2n + 3n + 5n
n"
2 + n sin n
lim
n"
n2 + 1
1 2 3
n
lim + +
n"
n n2 n3
"
n+1
lim n3 + n2 + 1
n"
 n
1
en = 1 + e
n
n
1
e := lim 1 + .
n"
n
e
2, 7182818285.
e
ln ln x := loge x
e
878 335588
e H" , e H" .
323 123456
e
(an) "
a
n
1
lim 1 + = e.
n"
an
(an)
"
a
n
1 1
lim 1 - = .
n"
an e
(an) -"
3n
1
lim 1 +
n"
n +
n2
1
lim 1 -
n"
n
2n+1
1
lim 1 -
n"
n2
2n+1
1
lim 1 +
n"
2n
n
3n + 1
lim
n"
3n + 4
2n+1
n - 1
lim
n"
n + 3
 (an), (bn)
an bn n n0
lim an = "
n"
lim bn = ".
n"
limn" [4n + (-1)n]
limn" [(2 cos n - 5)n2]
limn"[2n + sin n]
a + " = " a · " = "
a a
= 0 = "
" 0+
a" = 0 0+ a < 1 a" = " a > 1
"b = 0 b < 0 "b = " b > 0
a
= "
0+
(an), (bn)
lim an = a lim bn = 0 bn > 0 n " N
n" n"
an
lim = ".
n"
bn
2n - (n + 1)!
lim
n"
n + 1
2
n-n
2n2 + 1
lim
2
n"
"3n + 1
"
lim ( n + 3 - n)
n"
2 + n2 - n5
lim
n"
1 + n3
0 "
" - " 0 · " 1" "0 00
0 "
 (xn) (yn)
xn
lim xn = " lim yn = " lim
n" n" n"
yn
xn
xn = n2 yn = n lim = lim n = "
n" n"
yn
xn
xn = a · n a > 0 yn = n lim = lim a = a
n" n"
yn
xn 1
xn = n yn = n2 lim = lim = 0
n" n"
yn n
xn = (2 + (-1)n)n yn = n
xn
lim = lim (2 + (-1)n)
n" n"
yn