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
"
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