Granica ciągu
1. Oblicz granice następujących ciągów: (a) an = −2 − 3
b
c
d
n
n = n − 1
n
n = n+1
n2
n = n
2n
(b) an = ( 3n2+n−2 )2
b
− 3n
c
d
4n2+2n+7
n = cos n3
2n
6n+1
n = n2+sin n+3
4n2−n3
n = 3n+n2
2n2+1
√
√
√
(c) an =
n2 + n + 1 −
n2 − n + 1
bn = n2(n −
n2 + 1)
2. Oblicz granice wwykorzystując liczbę Eulera: (a) an = ( n+10 )n b
)n2+3
c
)15n
2n−1
n = (1 + 3
n2
n = ( 5n+2
5n+1
(b) an = ( n2 )n2
b
)5−2n
c
)3n−2
n2+1
n = ( n+4
n+3
n = (1 + 1
n
(c) an = [( 3n+2 )n · ( 5n+3 )n]
5n+2
3n+1
3. Oblicz granice wykorzystując twierdzenie o trzech ciągach:
√
√
√
(a) an = n n2 + 2n bn = n n10 + n!
cn = n+2 3n + 4n+1
√
(b) an = n 3 + sin n bn = 2n2+sin n1
c
4n2−3 cos n2
n = 2n sin n
3n+1
q
√
(c) a
1
n = n
+ 2 + 3 + 4
b
1 + 5n2 + 3n5
n
n2
n3
n4
n = n
(d) an =
1
+
1
+ . . . +
1
n2+1
n2+2
n2+n
1