n + 1
an = .
2n+1 + 1
{an}
n an+1 - an > 0
n an+1 - an < 0
n + 2 n + 1 (n + 2)(2n+1 + 1) - (n + 1)(2n+2 + 1)
an+1 - an = - = =
2n+2 + 1 2n+1 + 1 (2n+1 + 1)(2n+2 + 1)
n2n+1 + n + 2n+2 + 2 - n2n+2 - n - 2n+2 - 1
= =
(2n+1 + 1)(2n+2 + 1)
n2n+1 - n2n+1 · 2 + 1 1 - n2n+1
= = .
(2n+1 + 1)(2n+2 + 1) (2n+1 + 1)(2n+2 + 1)
an+1 - an < 0
1
an = .
(2n + 1)!
1
an+1 (2n+3)! (2n + 1)! (2n + 1)! 1
= = = = .
1
an (2n+1)! (2n + 3)! (2n + 1)!(2n + 2)(2n + 3) (2n + 2)(2n + 3)
an+1
an
0 < an < 1
{an}
"
n
" an = 5n + sin 3n
n
" L " R " n " N 5n + sin 3n < L
" " "
n
n n
5n + sin 3n 5n + 1 6n = 6,
L := 6
2n + 5
lim = 2.
n"
n + 1
 µ > 0 n0
|an - 2| < µ
2n + 5
- 2 < µ,
n + 1
3
< µ,
n + 1
3
< µ,
n + 1
n + 1 1
> ,
3 µ
3
n > - 1.
µ
3 3 3
- 1 + 1 - 1 + 1 > - 1
µ µ µ
3 2
n0 := - 1 + 1 µ = n0 = 7
µ 5
2n+5
n+1
2
2
5
" "
lim n 9n2 + 1 - 9n2 - 1 .
n"
n " [" - "]
a2 - b2
a - b = .
a + b
" "
" "
( 9n2 + 1)2 - ( 9n2 - 1)2
lim n 9n2 + 1 - 9n2 - 1 = lim n · " " =
n" n"
9n2 + 1 + 9n2 - 1
2n
9n2 + 1 - 9n2 + 1 2n
n
" "
= lim n · " " = lim " " = lim =
9n2+1 9n2-1
n" n" n"
9n2 + 1 + 9n2 - 1 9n2 + 1 + 9n2 - 1
+
n n
2 2 1
= lim = " " = .
n"
1 1 3
9 + 9
9 + + 9 -
n2 n2
"
n
lim 7n3 + 3n2 - 10n + 20.
n"
n
"
3 10 20
n
n
lim 7n3 + 3n2 - 10n + 20 = lim n3 7 + - + =
n" n"
n n2 n3
"
3 10 20
n
n
= lim n3 · 7 + - + =
n"
n n2 n3
"
3 10 20
n
n
= lim ( n)3 · 7 + - + .
n"
n n2 n3
 "
3 10 20
3
n
n
n - 1 7 + - + - 1.
n" n"
n n2 n3
"
"
3 10 20
n n
n
lim 7n3 + 3n2 - 10n + 20 = lim ( n)3 · lim 7 + - + = 1 · 1 = 1.
n" n" n"
n n2 n3
"
n
lim 7n + 30 · 2n + n3 + 4 cos n.
n"
n
"
2 n3 1
n
n
lim 7n + 30 · 2n + n3 + 4 cos n = lim 7n 1 + 30 · + + 4 · · cos n =
n" n"
7 7n 7n
n
"
2 n3 1
n
n
= lim 7n · 1 + 30 · + + 4 · · cos n =
n"
7 7n 7n
n
2 n3 1
n
= lim 7 · 1 + 30 · + + 4 · · cos n
n"
7 7n 7n
nk n3
lim = 0 k a - 0
n" n"
an 7n
n
1 2
- 0 - 0
n" n"
7n 7
{an} {bn}
1
lim anbn = 0 · cos n - 0
n" n"
7n
"
n
lim 7n + 30 · 2n + n3 + 4 cos n = 7
n"
"
n
lim 7n + 30 · 2n + n3 + 4 cos n.
n"
" " "
n n
n
7n d" 7n + 30 · 2n + n3 + 4 cos n d" 7n + 30 · 7n + 7n + 4 · 7n
" " " " "
n n n n
n
lim 7n + 30 · 7n + 7n + 4 · 7n = lim 36 · 7n = lim 36 · ·7n = lim 36 · 7 =
n" n" n" n"
1 · 7 = 7
2n
lim n · sin .
n"
1 + n2
[" · 0]
sin an
lim = 1
an0
an
2n 2n
sin sin
2n 2n 2n2
1+n2 1+n2
lim n · sin = lim n · · = lim · = 1 · 2 = 2,
2n 2n
n" n" n"
1 + n2 1 + n2 1 + n2
1+n2 1+n2
2n
- 0
n"
1 + n2
n!
an = .
nn
(n+1)!
an+1 (n + 1)! nn (n + 1)! nn
(n+1)n+1
lim = lim = lim = lim =
n!
n" n" n" n"
an (n + 1)n+1 n! n! (n + 1)n+1
nn
n
(n + 1) n! nn nn n
= lim = lim = lim =
n" n" n"
n! (n + 1)n(n + 1) (n + 1)n n + 1
-n n -1
n + 1 n + 1 1
= lim = lim = e-1 = < 1.
n" n"
n n e
an+1
lim < 1
n"
an
an+1
lim an = 0 lim > 1 lim an = "
n" n" n"
an
n!
lim an = lim = 0
n" n"
nn