-
MA , 1, 001
1. a) , ,
{xn} =
6n + 5
8n + 10
a = 3
4,
.
) , =13
100 N,
n > N ||xn a|| < .
2. limn
4n6 + 6n5 + 7n4 9n2 7n + 1
3n6 + 7n5 + 7n4 4n2 6n.
3. limn
2n + 9
n
2 2n + 72n 1
2
n + 6
22n + 2
2
n2 + 9n + 1.
4. limn
4 7n 2 5n + 3
2 7n 3 5n 2.
5. limn
4n
2 3 4n2 2 (5n + 4) .
6. limx2
2x2 x + 10
x3 8.
7. limx
(3x + 5)2
8x 4tg
3
x 4.
8. limn
8n 3
8n + 7
3 8n2
.
9.
f (x) =x2 + 13x + 42
||x + 6||(x2 2x 63).
10.
f (x) =
3x+1, x ( ; 2),
3x2 + 6x + 4, x [ 2; 2),
4x1, x (2; +) .
11. f (x) =x3 + 5x2 x 10
x2 6x + 9.
. 1 473 2012/2013 .
-
12. f (x) =3 2x 2 4x
2 2x + 4 4x.
. 2 473 2012/2013 .
-
MA , 1, 002
1. a) , ,
{xn} =
9n + 2
6n 4
a = 3
2,
.
) , =7
1000 N,
n > N ||xn a|| < .
2. limn
3n2 + 3n + 4
9n3 n2 + 2n 9.
3. limn
2n + 7
42n + 6
2n + 4
2n + 9
3n
4 + 712
.
4. limn
3n6 3 5n
5n6 4 5n.
5. limn
2 + 6n 9n2
3n 51 + 4n 2n2
7n 7.
6. limx4
x3 64
x2 16.
7. limx
(3x + 3)2
7x + 9tg
9
7x + 7.
8. limn
( 2n + 2)ln(2n 7) ln(2n 8) .
9.
f (x) =x2 4x 5
||x 5||(x2 + 9x + 8).
10.
f (x) =
2x4, x ( ; 2),
2x2 + 8x 7, x [2; 3],
8x3, x (3; +) .
11. f (x) =x3 + 2x2 + 3
x2 5x + 4.
. 3 473 2012/2013 .
-
12. f (x) =5 2x 5 3x
4 2x + 3 3x.
. 4 473 2012/2013 .
-
MA , 1, 003
1. a) , ,
{xn} =
3 5n 6
10 5n + 2
a =3
10,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
2n18 7n3 1
5n20 n4 2n 6.
3. limn
5n
2 8 sin6(5n + 4)7n + 3
n6n2 7n 8
.
4. limn
2 3n 2 4n 2
4 3n + 3 4n 3.
5. limn
16n
2 4n 2 4n .
6. limx2
4x2 + x + 14
x4 16.
7. limx
(x 6)3
6x 4sin2
1
9x + 7.
8. limn
( 6n + 1)ln(n + 3) ln(n 2) .
9. f (x) =x 1
||x2 3x + 2||.
10.
f (x) =
2x+1, x ( ; 8),
4x2 + 6x 1, x [ 8; 1),
3x+2, x ( 1; +) .
11. f (x) =x2 + 3x
x3 x.
12. f (x) =4 3x + 5 6x
3 3x + 2 6x.
. 5 473 2012/2013 .
-
MA , 1, 004
1. a) , ,
{xn} =
n 9
3n 7 a =
1
3,
.
) , =17
10000 N,
n > N ||xn a|| < .
2. limn
3n 7
8n2 + 9n + 9.
3. limn
6n + 6
5n 7 cos
5n + 82
5n + 4
3 .
4. limn
3n4 3 5n 4
4n6 + 3 3n 4.
5. limn
16n
2 4n + 5 4n .
6. limx2
6x2 x 22
3x2 + 5x + 2.
7. limx
7x4 + 7x 9
4x 7tg3
1
2x + 6.
8. limx
3x + 3
3x 2
3x3.
9. f (x) =||x 2||
x2 3x + 2.
10.
f (x) =
8x+2, x ( ; 2),
8x2 + 3x 6, x ( 2; 1),
2x+3, x ( 1; +) .
11. f (x) =x3 + 5x2 + 7x + 7
x2 + 4x + 4.
12. f (x) =3 2x 4 3x
2 2x + 2 3x.
. 6 473 2012/2013 .
-
MA , 1, 005
1. a) , ,
{xn} =
9 4n 6
5 4n a =
9
5,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
2n + 2
7.
3. limn
2n
3 4n2 + 7n
2 832
n6 n6
2n + 9 n 5
3 .
4. limn
3n3 + 2 + 2 7n
5 7n + 4n3 5.
5. limn
1 9n + 8n2
8n 96 + 7n + 2n2
2n + 7.
6. limx3
x4 81
3x2 6x 9.
7. limx
( x + 7)2
3x + 8sin
2
5x + 3.
8. limx1
xx+1x1 .
9. f (x) =||x + 2||
x2 2x 8.
10.
f (x) =
4x2, x ( ; 2),
2x2 8x 9, x ( 2; 4],
5x+7, x (4; +) .
11. f (x) =5x2 4x + 9
x2 + x 2.
12. f (x) =4 3x 2 5x
5 3x 2 5x.
. 7 473 2012/2013 .
-
MA , 1, 006
1. a) , ,
{xn} =
6 3n 6
3 3n 4 a = 2,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
9n3 + 6n2 + 3n 4
2n4 + 9n3 3n + 3.
3. limn
3n + 3
4 2n 4
2
2n + 1
3n
2 9 n + 6
3.
4. limn
5n2 + 5 4n + 2n
2n3 2 3n 3n + 4.
5. limn
n n
2 + 2n + 4 .
6. limx1
4x2 + 2x + 2
x3 1.
7. limx 5
x2 6x 5
2 tg(x + 5).
8. limx
3x5 + 9
3x5 + 7
8x5+4.
9. f (x) =x + 3
||x2 x 12||.
10.
f (x) =
8x+1, x ( ; 2),
2x2 + 4x + 8, x ( 2; 7],
4x7, x (7; +) .
11. f (x) =4x2 29x + 44
x2 7x + 12.
12. f (x) =5 2x 4 6x
4 2x + 4 6x.
. 8 473 2012/2013 .
-
MA , 1, 007
1. a) , ,
{xn} =
4 4n
2 4n 5 a = 2,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
7n 5 sin47n 8
9n 2 sin67n + 7.
3. limn
2n + 6
23n 6
3n + 7
3
3n + 1
2 3n + 6
22n + 9
2 .
4. limn
5 5n + 5 6n + 4
4 5n 3 6n 3.
5. limn
4 n 4n2
7n + 18 + 4n 6n2
n 2.
6. limx2
2x2 6x + 20
x5 32.
7. limx
5x4 9x + 6
6x + 8sin3
3
x + 2.
8. limn
3n 7
3n + 6
3n8
.
9.
f (x) =x2 7x + 10
||x 2||(x2 + x 30).
10.
f (x) =
6x+4, x ( ; 5),
6x2 + 6x 6, x ( 5; 1),
1x+2, x [ 1; +) .
11. f (x) =3x2 + 2x + 16
x2 4.
. 9 473 2012/2013 .
-
12. f (x) =3 3x 2 5x
5 3x 4 5x.
. 10 473 2012/2013 .
-
MA , 1, 008
1. a) , ,
{xn} =
7n + 4
3n + 3 a =
7
3,
.
) , =11
100000 N,
n > N ||xn a|| < .
2. limn
6n2 8
9n2 + 2n.
3. limn
n 8
2 n 4
22n 7
2n 6
n14 37
3n3 3
.
4. limn
4 7n 2 2n 2
2 2n 3 7n + 2.
5. limn
3n 9n
2 + 2n 2 .
6. limx 3
3x2 + 6x 9
x2 9.
7. limx 9
8 sin(x + 9)
x2 8x + 9.
8. limn
( 2n + 5)ln(4n + 1) ln(4n + 2) .
9.
f (x) =||x + 2||(x + 7)
x3 + 9x2 + 14x.
10.
f (x) =
4x+9, x ( ; 9),
6x2 + 7x 5, x ( 9; 1),
6x+2, x ( 1; +) .
11. f (x) =2x3 2x2 6x + 1
x2 2x + 1.
12. f (x) =4 2x 5 4x
4 2x + 3 4x.
. 11 473 2012/2013 .
-
MA , 1, 009
1. a) , ,
{xn} =
6 2n 2
10 2n + 3 a =
3
5,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
7n20 + 7n3 2n + 1
8n16 + 6n6 + 4.
3. limn
n
3 + 7n2 + 4n 6
2
n
4 7n3 + 3n2 72n 1
.
4. limn
5 5n + 4 3n
3 5n + 5 3n.
5. limn
6 8n + 8n2
7n + 33 7n + 8n2
7n 7.
6. limx 2
5x2 4x + 12
x5 + 32.
7. limx1
2 tg(x 1)
9x2 + 7x + 2.
8. limn
3n + 3
3n + 7
3n5
.
9. f (x) =x + 3
||x2 x 12||.
10.
f (x) =
2x5, x ( ; 5),
x2 + 9x + 9, x (5; 9),
9x8, x (9; +) .
11. f (x) =5x2 3x 7
x2 x 2.
12. f (x) =4 3x + 4 5x
2 3x + 5 5x.
. 12 473 2012/2013 .
-
MA , 1, 010
1. a) , ,
{xn} =
2 4n 9
8 4n + 10
a =1
4,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
3n 7 cos63n 2
9n + 2 sin6n 5.
3. limn
4n 6
3 2n 5
4
n12 + 26
n 923n + 5
2.
4. limn
2 6n 4n3 2
5 6n 3n3 4.
5. limn
3 4n + 7n2
7n 49 + 8n + 3n2
3n + 8.
6. limx1
2x2 3x + 1
x2 6x + 5.
7. limx
( x + 8)3
x + 7sin2
9
x + 2.
8. limx
9x4 + 9
9x4 + 7
8x41.
9.
f (x) =x2 5x 36
||x 9||(x2 + 12x + 32).
10.
f (x) =
7x5, x ( ; 4),
2x2 8x + 7, x (4; 9),
2x9, x (9; +) .
11. f (x) =x3 2x2 5x + 6
x2 5x + 6.
. 13 473 2012/2013 .
-
12. f (x) =4 3x 4 4x
5 3x + 2 4x.
. 14 473 2012/2013 .
-
MA , 1, 011
1. a) , ,
{xn} =
6 5n + 2
3 5n 8
a = 2,
.
) , =1
1000000000 N,
n > N ||xn a|| < .
2. limn
4n2 + 2n + 5 + 9n2 4n + 2
4n + 4.
3. limn
n3n 2n
6 116
n
4 514 3n + 8
4n + 2
.
4. limn
5n6 + 3 4n
2n7 + 4 3n + 1.
5. limn
2 4n + 2n2
2n 78 6n 4n2
9n 8.
6. limx1
3x2 + 6x 9
3x2 4x + 1.
7. limx 2
3x2 + 4x 4
8 tg(x + 2).
8. limx
3x + 4
3x 2
x+2.
9. f (x) =||x||(x + 4)
x3 + x2 12x.
10.
f (x) =
3x6, x ( ; 5),
x2 + 5x + 3, x [5; 6],
3x5, x (6; +) .
11. f (x) =x3 + 7x 6
x2 3x + 2.
12. f (x) =3 3x 4 5x
2 3x 3 5x.
. 15 473 2012/2013 .
-
MA , 1, 012
1. a) , ,
{xn} =
5n + 7
3n 8
a = 5
3,
.
) , =11
100000 N,
n > N ||xn a|| < .
2. limn
6n + 8 cos47n + 8
3n + 8 cos4n + 2.
3. limn
n
3 2n2 + 5n + 9n
2 + 512
n + 2
2n
2 4n + 1 n2 + 1
.
4. limn
3 6n 2 5n + 4
4 6n + 2 5n + 2.
5. limn
7 + 4n + 2n2
6n + 47 7n + 2n2
6n 8.
6. limx 1
5x2 6x 11
x2 4x 3.
7. limx 2
8 tg(x + 2)
5x2 + 8x 4.
8. limx0
4 + 5x
4 3x
3 x7x .
9. f (x) =x 2
||x2 + 2x 8||.
10.
f (x) =
6x5, x ( ; 1),
5x2 + 4x 2, x ( 1; 6],
8x6, x (6; +) .
11. f (x) =3x2 + 6x
x3 + 2x2 3x.
12. f (x) =4 3x + 5 4x
2 3x + 4 4x.
. 16 473 2012/2013 .
-
MA , 1, 013
1. a) , ,
{xn} =
7n + 1
n 6 a = 7,
.
) , =11
10000 N,
n > N ||xn a|| < .
2. limn
7n 6 sin35n + 6
7n 8 cos7n 1.
3. limn
4n
2 93
3n
2 36n
2 32 .
4. limn
4n3 3 4n + 5n
5n5 2 3n + 2.
5. limn
7 5n + 8n2
5n 53 7n 3n2
3n 7.
6. limx 2
5x + 74 8
6x 12.
7. limx1
4 sin(x 1)
7x2 + 5x + 2.
8. limx1
x9x+71 x .
9. f (x) =||x + 1||
x2 x 2.
10.
f (x) =
3x3, x ( ; 2],
2x2 4x + 3, x (2; 3),
9x2, x (3; +) .
11. f (x) =2x2 + 6x
x3 + x2 12x.
12. f (x) =2 3x + 2 5x
2 3x 3 5x.
. 17 473 2012/2013 .
-
MA , 1, 014
1. a) , ,
{xn} =
8 4n 5
7 4n + 2 a =
8
7,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
7n + 4 sin6n 8
6n + 4 sin5n + 7.
3. limn
3n 9
6
9n
2 + 67n
2 52 .
4. limn
2 3n + 3 6n 2
5 6n + 4 3n 4.
5. limn
2 + 7n + 7n2
7n 73 5n + 8n2
8n 5.
6. limx0
7x + 9 3
5x + 25 5.
7. limx
2x4 4x + 1
5x 2sin3
4
3x 6.
8. limx1
x7x+31 x .
9.
f (x) =x2 13x + 36
||x 4||(x2 16x + 63).
10.
f (x) =
3x4, x ( ; 3),
2x2 5x 6, x [3; 4],
6x3, x (4; +) .
11. f (x) =3x2 + 11x + 10
x2 3x 4.
. 18 473 2012/2013 .
-
12. f (x) =2 3x + 4 4x
4 3x 3 4x.
. 19 473 2012/2013 .
-
MA , 1, 015
1. a) , ,
{xn} =
8 4n + 5
2 4n
a = 4,
.
) , =1
1000 N,
n > N ||xn a|| < .
2. limn
3n4 + 8n3 + 3
8n18 + 4n8 6.
3. limn
2n 3
2 3n
3 + 7n2 8n + 6 3n 1
22n + 3
2 2n + 9
.
4. limn
3n3 5 6n + 2n 3
2n4 4 5n.
5. limn
4n
2 4 4n2 + 3 (3n 1) .
6. limx 2
x3 + 8
x5 + 32.
7. limx
x2 7x 7
9x 9sin
7
4x + 4.
8. limn
6n + 5
6n + 2
5n+2
.
9. f (x) =||x 5||
x2 7x + 10.
10.
f (x) =
2x+2, x ( ; 3),
2x2 + 5x 5, x ( 3; 1],
6x+6, x ( 1; +) .
11. f (x) =x3 + 6x2 + 6x 4
x2 + 5x + 6.
12. f (x) =4 4x + 4 5x
5 4x 2 5x.
. 20 473 2012/2013 .
-
MA , 1, 016
1. a) , ,
{xn} =
3n 2
3n + 4
a = 1,
.
) , =3
1000 N,
n > N ||xn a|| < .
2. limn
3n14 + 8n9 6n 3
7n14n4 + 5.
3. limn
9n
3 + 6n2 + 2n 25n
2 + 8 cos(3n + 3)2n 7
.
4. limn
3 5n 5 3n
5 5n + 4 3n.
5. limn
5 7n + 7n2
5n 44 + 8n + 7n2
5n + 4.
6. limx 2
x3 + 8
4x2 + 2x 12.
7. limx
( 3x + 2)2
4x 7sin
1
4x 2.
8. limn
3n 1
3n + 2
4 3n2
.
9. f (x) =x 2
||x2 3x + 2||.
10.
f (x) =
5x+2, x ( ; 3),
x2 4x + 2, x ( 3; 2),
6x+3, x [ 2; +) .
11. f (x) =4x3 3x2
x2 25.
12. f (x) =2 2x 3 4x
3 2x + 2 4x.
. 21 473 2012/2013 .
-
MA , 1, 017
1. a) , ,
{xn} =
4 4n + 7
9 4n + 6 a =
4
9,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
7n2 2
6n4 4n3 8n2 + 5.
3. limn
3n
2 + 9 9n
2 + 72
7n
6 2.
4. limn
2 2n + 5n3 2
2n3 + 4 2n 3.
5. limn
2n 4n
2 + 2n 4 .
6. limx 2
x4 16
2x2 5x 18.
7. limx0
x tg8x 9 sin( 6x)
7x2cos5x+ 8 sin( 7x 2x2)
.
8. limn
5n + 9
5n + 7
3 5n8
.
9.
f (x) =x2 3x 54
||x + 6||(x2 17x + 72).
10.
f (x) =
4x+4, x ( ; 3),
2x2 + 4x 2, x [ 3; 2),
2x+3, x [ 2; +) .
11. f (x) =2x3 + 6x2 + 2x 5
x2 + 4x + 4.
. 22 473 2012/2013 .
-
12. f (x) =5 2x + 4 4x
4 2x + 3 4x.
. 23 473 2012/2013 .
-
MA , 1, 018
1. a) , ,
{xn} =
9 2n 2
5 2n + 8 a =
9
5,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
16n2 + 3n 1 + 9n2 + 2n + 3
4n2 + 4n 2 n2 n 4.
3. limn
4n 6
2 7n
2 + 22
n 1
6
.
4. limn
3n4 + 4 5n
4n6 2 4n 2n.
5. limn
4
n2 + 4n 1 n2 + 3n 2.
6. limx 2
4x2 5x + 6
x2 3x 2.
7. limx0
4 sin x + x tg5x
3 sin( 2x) 8x2sin7x
.
8. limx1
x5x71 x .
9.
f (x) =x2 14x + 45
||x 5||(x2 3x 54).
10.
f (x) =
7x+1, x ( ; 2],
2x2 + 6x 3, x ( 2; 1],
7x+2, x ( 1; +) .
11. f (x) =x3 + x2 + x 2
x2 + 2x + 1.
. 24 473 2012/2013 .
-
12. f (x) =3 2x + 4 6x
4 2x + 3 6x.
. 25 473 2012/2013 .
-
MA , 1, 019
1. a) , ,
{xn} =
4n 5
n + 7 a = 4,
.
) , =7
1000 N,
n > N ||xn a|| < .
2. limn
7n5 6n3 8
2n18 2n + 2.
3. limn
7n 9 sin
6(5n 7)2 3n 8
7n + 4
3 .
4. limn
2n3 5 6n
3n6 2 3n + 2n.
5. limn
3n 9n
2 5n + 1 .
6. limx 2
x2 4x 12
4x2 + 4x 8.
7. limx
2x5 8x 3
7x 2sin4
1
3x 1.
8. limn
(7n + 2)ln(4n 5) ln(4n + 3) .
9.
f (x) =x2 x 12
||x + 3||(x2 11x + 28).
10.
f (x) =
9x+2, x ( ; 3],
2x2 x + 6, x ( 3; 1),
5x+2, x [ 1; +) .
11. f (x) =3x2 + 20x + 28
x2 + 6x + 8.
12. f (x) =2 3x 4 5x
4 3x + 5 5x.
. 26 473 2012/2013 .
-
MA , 1, 020
1. a) , ,
{xn} =
n 2
11n + 7
a = 1
11,
.
) , =11
1000 N,
n > N ||xn a|| < .
2. limn
7
9n + 6.
3. limn
2n
2 + 2n 3n
2 + 432
n2 + 5 3n + 24 .
4. limn
3n4 3 5n
2n6 + 3 3n n 3.
5. limn
3
25n2 4n + 2 5n 3.
6. limx2
6x2 2x 20
6x2 + 4x + 16.
7. limx0
tg(4x2) 6 sin( x)
3 sin( 2x + x2) 6x2arctg8x
.
8. limx1
x5x+21 x .
9. f (x) =x 2
||x2 + 2x 8||.
10.
f (x) =
8x7, x ( ; 1],
8x2 + 3x 6, x ( 1; 7],
2x+6, x (7; +) .
11. f (x) =2x3 + 3x2 4x 4
x2 2x + 1.
12. f (x) =4 4x + 2 6x
5 4x + 4 6x.
. 27 473 2012/2013 .
-
MA , 1, 021
1. a) , ,
{xn} =
3 3n 6
2 3n 8 a =
3
2,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
4n2 2n + 4 + n2 + 2n 3
2n 1.
3. limn
n n 7 3n
3 3
n7 17
3n 22n
6 + 416
.
4. limn
4 6n + 5n3 3
3n3 4 6n + 3.
5. limn
n n
2 4n + 2 .
6. limx3
2x2 + 5x + 3
3x2 + 5x + 12.
7. limx0
tg(8x2) 3 tg4x
7 sin( 8x) + 3x2sin8x
.
8. limn
7n 5
7n 3
9n9
.
9.
f (x) =x2 2x 35
||x 7||(x2 4x 45).
10.
f (x) =
2x+3, x ( ; 4),
2x2 + 6x 6, x ( 4; 3],
6x+4, x ( 3; +) .
11. f (x) =3x3 + 4x2 9x 10
x2 + x 2.
. 28 473 2012/2013 .
-
12. f (x) =3 2x 5 3x
4 2x 4 3x.
. 29 473 2012/2013 .
-
MA , 1, 022
1. a) , ,
{xn} =
2 4n 4
9 4n + 2 a =
2
9,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
1
5n4 + 8.
3. limn
nn 92n
2 932
3n 1
2n 8
3 .
4. limn
3n5 + 2 5n 3
3n6 3 2n 5n + 1.
5. limn
25n
2 3n + 3 5n .
6. limx9
4x + 36
3x + 54 9.
7. limx 1
9 tg(x + 1)
2x2 + x + 3.
8. limx0
1 + 2x
1 x
1arcsin8x .
9.
f (x) =x2 + 3x 18
||x + 6||(x2 9x + 18).
10.
f (x) =
3x+1, x ( ; 2),
x2 2x + 3, x [ 2; 2),
5x1, x [2; +) .
11. f (x) =2x3
x2 16.
. 30 473 2012/2013 .
-
12. f (x) =4 3x 5 5x
2 3x 5 5x.
. 31 473 2012/2013 .
-
MA , 1, 023
1. a) , ,
{xn} =
4 4n
9 4n 2 a =
4
9,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
5n14 + 6n10 4
7n10 7.
3. limn
6n + 7
3
6n 4 sin n 3
29n 5
.
4. limn
2 6n 3n6 + 5
2 6n + 4n6 + 4.
5. limn
4n
2 n + 4 2n .
6. limx 1
x3 + 1
x4 1.
7. limx0
6 tg( 7x) 7x2arctg1x
x sin6x 6 tg4x.
8. limx0
4 x
4 2x
3+3x4x .
9.
f (x) =x2 + x 56
||x 7||(x2 + 10x + 16).
10.
f (x) =
2x6, x ( ; 6),
9x2 4x 7, x (6; 7),
4x+8, x (7; +) .
11. f (x) =2x3
x2 16.
. 32 473 2012/2013 .
-
12. f (x) =5 3x + 3 6x
4 3x 2 6x.
. 33 473 2012/2013 .
-
MA , 1, 024
1. a) , ,
{xn} =
7 3n 4
8 3n + 11
a =7
8,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
3n 4 sin55n 7
6n 7 sin3n 4.
3. limn
n2 4 2n 6n + 1
n
2 412 2n 6
2
.
4. limn
3n4 + 2 7n + 4n + 3
2n5 3 2n 3.
5. limn
1 + 4n + 2n2
2n + 81 2n + 3n2
3n + 7.
6. limx7
7x 24 5
4x 28.
7. limx
4x3
3x + 5sin2
1
4x + 7.
8. limx0
1 + x
1 + 4x
3+4x2x .
9.
f (x) =x2 + 2x 24
||x 4||(x2 + 10x + 24).
10.
f (x) =
4x2, x ( ; 2),
x2 + 9x + 8, x [2; 9],
4x9, x (9; +) .
11. f (x) =x3 4x2 x + 4
x2 4x + 3.
. 34 473 2012/2013 .
-
12. f (x) =2 3x 3 4x
3 3x 4 4x.
. 35 473 2012/2013 .
-
MA , 1, 025
1. a) , ,
{xn} =
6 2n 2
7 2n 8 a =
6
7,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
2n3 + 7 cos8n 9
6n3 3 cos6n + 9.
3. limn
2n + 2
29n
2 52
n
6 + 9.
4. limn
4 6n + 2n7 4
5n7 2 6n 5.
5. limn
1 + 8n + 9n2
9n + 88 3n + 3n2
3n 3.
6. limx0
2x + 49 7
3x.
7. limx
4x2 7x + 9
3x 7tg
5
8x 2.
8. limn
4n 5
4n + 4
9n5
.
9.
f (x) =||x||(x + 6)
x3 + 10x2 + 24x.
10.
f (x) =
2x9, x ( ; 2),
4x2 + 5x 4, x [ 2; 9],
3x9, x (9; +) .
11. f (x) =x3 + 5x2 6x 8
x2 6x + 9.
. 36 473 2012/2013 .
-
12. f (x) =5 3x + 3 4x
2 3x + 3 4x.
. 37 473 2012/2013 .
-
MA , 1, 026
1. a) , ,
{xn} =
6n + 1
9n 5 a =
2
3,
.
) , =17
100000 N,
n > N ||xn a|| < .
2. limn
3n10 + 8
7n18 5n + 3.
3. limn
n
2 + 612 2n
4 6n2 8n + 3n 8
4
n2 2n + 4.
4. limn
5n5 + 4 4n 5n + 5
2n6 4 3n.
5. limn
8 + 5n + 3n2
3n + 52 + 9n + n2
n + 9.
6. limx2
x4 16
5x2 + 2x + 16.
7. limx
5x2 3x 6
x 8sin
4
3x 9.
8. limn
8n 7
8n + 2
8n+6
.
9.
f (x) =x2 + x 12
||x + 4||(x2 + 4x 21).
10.
f (x) =
3x4, x ( ; 3],
2x2 + 5x + 6, x (3; 4),
6x3, x (4; +) .
11. f (x) =4x3 5x2 + 4
x2 + 2x + 1.
. 38 473 2012/2013 .
-
12. f (x) =4 3x 4 4x
3 3x + 2 4x.
. 39 473 2012/2013 .
-
MA , 1, 027
1. a) , ,
{xn} =
5n + 9
3n + 2
a = 5
3,
.
) , =1
1000 N,
n > N ||xn a|| < .
2. limn
6n + 2 cos35n 2
7n + 3 cos26n 7.
3. limn
2n
2 + 2n 3n
2 432
4n 4
2n
2 832
.
4. limn
4n3 + 5 6n
3n6 + 3 5n n 2.
5. limn
1 8n + 3n2
3n 82 + 9n + 7n2
7n + 9.
6. limx3
2x2 + 5x + 3
x2 x + 12.
7. limx1
4x2 + 3x 7
3 sin(x 1).
8. limn
(7n 2)ln(5n 9) ln(5n + 4) .
9. f (x) =||x 2||(x 3)
x3 5x2 + 6x.
10.
f (x) =
4x3, x ( ; 1],
6x2 8x 1, x ( 1; 6),
3x6, x (6; +) .
11. f (x) =x2 2x
x3 16x.
12. f (x) =2 2x + 4 3x
4 2x 4 3x.
. 40 473 2012/2013 .
-
MA , 1, 028
1. a) , ,
{xn} =
8n 7
9n + 3
a = 8
9,
.
) , =17
10000 N,
n > N ||xn a|| < .
2. limn
8
8n6 7n4 + 3n3 + 7n2 4n + 3.
3. limn
4n
2 + 4n + 22n + 5 sin
2( 8n + 7) 5n + 3
3 .
4. limn
3n7 2 4n
4n5 3 6n 2.
5. limn
8 + 2n + 5n2
7n + 76 + 8n + 8n2
4n 5.
6. limx 2
3x2 + 5x 2
2x2 + 5x + 2.
7. limx
(4x + 8)2
5x + 9sin
1
3x 3.
8. limn
2n 3
2n 2
n1
.
9.
f (x) =x2 + 5x 24
||x + 8||(x2 + 2x 15).
10.
f (x) =
9x+2, x ( ; 3],
2x2 4x 3, x ( 3; 2],
3x+3, x ( 2; +) .
11. f (x) =3x3 + 2x2 + 3
x2 + 2x 8.
. 41 473 2012/2013 .
-
12. f (x) =2 2x 4 4x
4 2x + 4 4x.
. 42 473 2012/2013 .
-
MA , 1, 029
1. a) , ,
{xn} =
7n + 1
9n + 2
a = 7
9,
.
) , =11
10000 N,
n > N ||xn a|| < .
2. limn
8n 6 sin53n 2
9n 9 cos7n + 7.
3. limn
2n 4
2 n + 6
2n
6 + 816
2n 4
3n + 8
3n
2 + 512
.
4. limn
2 5n 3 2n 4
3 5n 2 2n 2.
5. limn
2 + 8n + 5n2
5n + 87 + 3n + 4n2
4n + 3.
6. limx 4
8x + 113 9
8x + 32.
7. limx
2x3
9x 7tg2
8
3x + 5.
8. limx
x 4
x + 2
5x1.
9. f (x) =||x||(x + 1)
x3 x2 2x.
10.
f (x) =
6x4, x ( ; 8),
3x2 2x 6, x ( 8; 1),
1x+2, x ( 1; +) .
11. f (x) =x3 + 2x2 + 6x + 3
x2 2x 3.
12. f (x) =3 3x 5 4x
3 3x + 4 4x.
. 43 473 2012/2013 .
-
MA , 1, 030
1. a) , ,
{xn} =
8 2n + 10
9 2n 5
a = 8
9,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
3
6n3 + 7n2 7.
3. limn
n3n 62n
7 717
n5 3n5
n 2n
4 412
.
4. limn
3n2 + 4 5n + 1
2n7 + 3 3n 4n + 5.
5. limn
4 + 6n + 7n2
4n + 37 4n + 7n2
6n 1.
6. limx 1
4x2 + 4x + 8
5x2 + 3x 2.
7. limx0
6x2arctg4x
3 tg( 5x + x
2)
2 sin( 3x) + tg(8x2).
8. limx1
xx+51 x .
9. f (x) =x 3
||x2 5x + 6||.
10.
f (x) =
9x+7, x ( ; 7),
2x2 8x + 7, x [ 7; 2),
3x+1, x (2; +) .
11. f (x) =2x2 6x 4
x3 + 5x2 + 6x.
. 44 473 2012/2013 .
-
12. f (x) =3 4x 4 5x
2 4x + 3 5x.
. 45 473 2012/2013 .
-
MA , 1, 031
1. a) , ,
{xn} =
7n 8
9n 6
a = 7
9,
.
) , =7
100 N,
n > N ||xn a|| < .
2. limn
n3 + 5n 9
5n4 + 6n3 + 1.
3. limn
2n + 4
22n 4
2 3n 4
n 7
3n
2 3n
8 418
.
4. limn
4n7 4 5 2n
2 2n + 5n7 + 5.
5. limn
4n
2 + 5 4n2 + 3 (2n + 1) .
6. limx3
8x + 24
9x + 37 8.
7. limx
7x5 + 7x 7
5x 3tg4
1
x 2.
8. limx
3x 3
3x 2
2x5.
9.
f (x) =||x + 7||(x + 5)
x3 + 12x2 + 35x.
10.
f (x) =
6x4, x ( ; 3],
x2 + 4x + 3, x (3; 4),
3x3, x (4; +) .
11. f (x) =x2 + 5x + 36
x2 x 20.
. 46 473 2012/2013 .
-
12. f (x) =4 3x 2 4x
4 3x 3 4x.
. 47 473 2012/2013 .
-
MA , 1, 032
1. a) , ,
{xn} =
5n + 9
9n 6 a =
5
9,
.
) , =11
100 N,
n > N ||xn a|| < .
2. limn
2n3 8
5n2 7.
3. limn
8n
2 + 88n
2 72
n
2 + 53 .
4. limn
3 7n + 3 3n 5
4 7n + 4 3n 4.
5. limn
9n
2 + 4n 5 3n .
6. limx9
2x + 67 7
x 9.
7. limx
8x3 6x2
7x 4tg2
2
9x 9.
8. limx
7x7 + 1
7x7 + 7
3x76.
9.
f (x) =||x||(x + 2)
x3 5x2 14x.
10.
f (x) =
3x+1, x ( ; 1),
3x2 + 5x 2, x ( 1; 8),
8x+9, x (8; +) .
11. f (x) =3x3 + 10x2 + x 5
x2 + 4x + 4.
. 48 473 2012/2013 .
-
12. f (x) =5 3x 2 5x
3 3x 2 5x.
. 49 473 2012/2013 .
-
MA , 1, 033
1. a) , ,
{xn} =
10 3n + 6
8 3n + 7
a =5
4,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
8n2 + 7
6.
3. limn
6n
2 + 85n
2 + 43n
2 2n + 3
6 .
4. limn
2n5 2 4n 4n 3
4n7 3 3n + 3n.
5. limn
5 + 2n + 8n2
n + 96 7n + 9n2
7n 1.
6. limx 1
2x2 + 2x + 4
2x2 + 5x + 7.
7. limx
x3
9x + 1sin2
3
7x + 3.
8. limn
3n 2
3n + 5
9n1
.
9.
f (x) =||x||(x 2)
x3 + 5x2 14x.
10.
f (x) =
7x+5, x ( ; 5),
x2 + 3x + 2, x ( 5; 5),
8x4, x (5; +) .
11. f (x) =3x2 6x 11
x2 x 2.
. 50 473 2012/2013 .
-
12. f (x) =2 2x 3 4x
5 2x + 2 4x.
. 51 473 2012/2013 .
-
MA , 1, 034
1. a) , ,
{xn} =
5 3n 3
7 3n + 7 a =
5
7,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
2
n2 6.
3. limn
n3n + 8 n + 9
n + 7
2
n
4 614 n 6
22n + 4
2.
4. limn
2 6n 5 7n 3
4 7n 5 6n + 2.
5. limn
n
2 4n + 2 n .
6. limx 4
x2 + 3x 4
x2 16.
7. limx
5x4 + 8x 6
6x + 1tg3
1
2x 3.
8. limn
5n + 9
5n 5
2n+7
.
9. f (x) =||x + 1||(x 6)
x3 5x2 6x.
10.
f (x) =
3x+6, x ( ; 6),
7x2 3x + 9, x [ 6; 8],
3x4, x (8; +) .
11. f (x) =x3 2x2 10x 4
x2 + x 2.
12. f (x) =2 3x 4 4x
5 3x 2 4x.
. 52 473 2012/2013 .
-
MA , 1, 035
1. a) , ,
{xn} =
9 3n 5
3 3n 8
a = 3,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
7n14 5
3n14 + 6n 8.
3. limn
9n
3 5n2 8n 2n8n 3
7n 9 sin n + 7
.
4. limn
4 7n + 4 2n + 2
5 7n + 2 2n 3.
5. limn
25n
2 + 5n 2 5n .
6. limx3
x3 27
x4 81.
7. limx0
6 tg( x) + 8x2sin3x
3 sin( 3x) + sin(2x2).
8. limx1
x5x+91 x .
9. f (x) =x + 2
||x2 x 6||.
10.
f (x) =
1x+1, x ( ; 2),
2x2 + 6x + 5, x ( 2; 1),
1x+2, x [ 1; +) .
11. f (x) =5x2 + 5x 8
x2 1.
12. f (x) =5 3x 2 5x
5 3x + 2 5x.
. 53 473 2012/2013 .
-
MA , 1, 036
1. a) , ,
{xn} =
7 3n 5
11 3n a =
7
11,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
3n5 6n4 + 3
7n18 + n4 2.
3. limn
3n 9
2 9n
2 + 72
3n + 3
6 .
4. limn
4 4n + 4n4 4
3 4n + 5n4 3.
5. limn
n n
2 2n + 2 .
6. limx3
5x + 66 9
x 3.
7. limx
(x + 1)3
8x + 8tg2
5
6x + 9.
8. limn
5n + 2
5n 1
3n+4
.
9. f (x) =x + 1
||x2 x 2||.
10.
f (x) =
2x4, x ( ; 8),
8x2 + 4x 6, x [ 8; 1),
6x+4, x [ 1; +) .
11. f (x) =3x2 10x + 9
x2 5x + 4.
12. f (x) =3 4x 4 5x
3 4x 2 5x.
. 54 473 2012/2013 .
-
MA , 1, 037
1. a) , ,
{xn} =
4 3n 2
2 3n + 3 a = 2,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
5n14 8
2n14 + 9n9 8n5 8.
3. limn
4n 1
4n 6
n + 7
n2 + 3 2n4 + 3n3 + 4n2 1
.
4. limn
3 5n 4n3 + 5
4n3 + 2 5n 2.
5. limn
8 + 9n + 3n2
3n + 94 + 4n + 6n2
6n + 4.
6. limx0
8x + 64 8
3x + 9 3.
7. limx
7x5 3x + 7
6x + 2sin4
3
x + 1.
8. limn
3n 4
3n 5
7n2
.
9. f (x) =x + 4
||x2 + x 12||.
10.
f (x) =
2x7, x ( ; 1),
x2 + 5x 3, x ( 1; 4),
2x2, x (4; +) .
11. f (x) =5x2 + 23x 12
x2 5x + 4.
12. f (x) =4 2x 3 6x
5 2x + 4 6x.
. 55 473 2012/2013 .
-
MA , 1, 038
1. a) , ,
{xn} =
11n + 10
6n + 10
a =11
6,
.
) , =17
1000 N,
n > N ||xn a|| < .
2. limn
5n6 9n3 + 5n2 + 2
4n7 6n + 4.
3. limn
4n
3 + n2 + 5n + 7n
4 + 412
3n 7
23n 7
3 .
4. limn
4n4 + 3 + 2 3n
4 3n 3n4 + 5.
5. limn
2n 4n
2 + 2n 3 .
6. limx0
6x + 25 5
7x + 36 6.
7. limx0
8 sin( 7x) + tg( 9x2)
6x2cos9x
+ 3 tg( 2x)
.
8. limn
( 2n 3)ln(2n 7) ln(2n + 2) .
9. f (x) =x 1
||x2 4x + 3||.
10.
f (x) =
2x+4, x ( ; 3),
2x2 + 4x 4, x [ 3; 1],
6x+2, x ( 1; +) .
11. f (x) =4x3 + x2 2
x2 4x + 3.
12. f (x) =2 3x 2 4x
3 3x 4 4x.
. 56 473 2012/2013 .
-
MA , 1, 039
1. a) , ,
{xn} =
6 2n 4
10 2n 6 a =
3
5,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
6n 6 cos6n + 4
5n 6 cos76n + 3.
3. limn
n3n 42
5n + 7
6n + 3 cos n + 4
2 .
4. limn
5 5n + 3n7 3
5 5n 2n7 + 4.
5. limn
4n 16n
2 + 3n + 2 .
6. limx2
5x2 + 2x 24
x4 16.
7. limx2
8 sin(x 2)
3x2 4x 4.
8. limn
8n + 3
8n 8
9n+3
.
9.
f (x) =x2 8x + 15
||x 5||(x2 12x + 27).
10.
f (x) =
5x+3, x ( ; 4],
2x2 6x + 3, x ( 4; 1),
7x , x [ 1; +) .
11. f (x) =x3 2x2 + 7x + 7
x2 + 4x + 4.
. 57 473 2012/2013 .
-
12. f (x) =2 2x + 2 6x
2 2x + 4 6x.
. 58 473 2012/2013 .
-
MA , 1, 040
1. a) , ,
{xn} =
2 5n 6
10 5n + 6 a =
1
5,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
3
7n7 + 4n5 5n4 + 9n2 + n + 6.
3. limn
2n 2
2n
2 + 132
n
2 612 n
4 + 3n3 + 2n2 + 3
.
4. limn
3n3 + 2 + 3 3n
2 3n 4n3 + 5.
5. limn
3 + n 9n2
2n 81 6n n2
6n + 4.
6. limx1
3x2 3x + 6
2x2 6x + 4.
7. limx
x2 7x 9
9x 4sin
1
2x + 5.
8. limn
4n + 1
4n + 8
4n7
.
9.
f (x) =x2 + x 6
||x + 3||(x2 9x + 14).
10.
f (x) =
2x+2, x ( ; 3],
x2 4x 1, x ( 3; 1),
2x+2, x [ 1; +) .
11. f (x) =3x2 + 4x 55
x2 25.
. 59 473 2012/2013 .
-
12. f (x) =2 5x 4 6x
3 5x 5 6x.
. 60 473 2012/2013 .
-
MA , 1, 041
1. a) , ,
{xn} =
8 3n 4
7 3n + 5 a =
8
7,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
2n + 3
8n.
3. limn
5n
3 + 5n2 3n 5 8n
2 + 3 sin( 7n 5) 5n + 5
.
4. limn
3 2n + 4 5n 5
4 2n 2 5n 3.
5. limn
7 n 7n2
3n 89 2n 9n2
4n + 2.
6. limx 2
2x + 45 7
2x + 60 8.
7. limx0
x sin3x + 7 tg( 5x)
6x2arctg8x
4 tg(9x + 2x
2).
8. limx
2x2 8
2x2 + 4
5x23.
9. f (x) =||x + 5||
x2 + 3x 10.
10.
f (x) =
3x+4, x ( ; 5],
3x2 + 6x 4, x ( 5; 2),
2x1, x [2; +) .
11. f (x) =4x3 + 9x2 2x 6
x2 x 2.
12. f (x) =3 5x + 3 6x
3 5x + 2 6x.
. 61 473 2012/2013 .
-
MA , 1, 042
1. a) , ,
{xn} =
n + 2
2n + 9 a =
1
2,
.
) , =1
100 N,
n > N ||xn a|| < .
2. limn
5n12 + 7n 1
6n6 2n2 + 3.
3. limn
2n 8
3
n2 1
n6 + 3n6 3n + 3n
4 912
.
4. limn
3 5n 2 2n
4 5n 5 2n.
5. limn
9 8n + 6n2
6n 86 + 7n + n2
n + 7.
6. limx2
3x2 + x + 10
x3 8.
7. limx
( 7x 6)2
6x + 1sin
1
4x 5.
8. limn
8n 9
8n + 4
8n5
.
9.
f (x) =x2 + 13x + 36
||x + 4||(x2 + 16x + 63).
10.
f (x) =
3x+2, x ( ; 3],
x2 3x + 3, x ( 3; 1],
5x , x ( 1; +) .
11. f (x) =x2 5x 14
x2 + 7x + 12.
. 62 473 2012/2013 .
-
12. f (x) =2 3x + 3 6x
4 3x 4 6x.
. 63 473 2012/2013 .
-
MA , 1, 043
1. a) , ,
{xn} =
7n + 6
2n 7
a = 7
2,
.
) , =7
100 N,
n > N ||xn a|| < .
2. limn
3n18 + 3n + 7
8n9 + 8 + 5n12 + 3n 5.
3. limn
n 1
2n
2 5 2n + 7
4
4n + 2
3 n
4 n2 + 4n 7
.
4. limn
4n3 + 3 + 3 3n
4 3n 2n3 2.
5. limn
n
2 + n 2 n .
6. limx 1
x3 + 1
3x2 + 2x 1.
7. limx0
tg(6x2) + 3 sin7x
2x2sin5x
+ 8 sin(2x 3x2)
.
8. limn
7n 5
7n 3
8n+6
.
9.
f (x) =x2 + x 56
||x + 8||(x2 5x 14).
10.
f (x) =
8x+5, x ( ; 7),
4x2 9x 2, x [ 7; 6),
5x6, x (6; +) .
11. f (x) =2x2 x 7
x2 x 2.
. 64 473 2012/2013 .
-
12. f (x) =3 2x + 5 5x
3 2x + 4 5x.
. 65 473 2012/2013 .
-
MA , 1, 044
1. a) , ,
{xn} =
7 4n + 6
3 4n + 6 a =
7
3,
.
) , =1
1000 N,
n > N ||xn a|| < .
2. limn
7n2 + 4n + 6
4n3 + 8n 3.
3. limn
3n
6 8 8n
2 56n
2 + 54n
2 + 3.
4. limn
2 4n 2 5n 2
3 5n + 3 4n + 3.
5. limn
9n
2 4n 3 3n .
6. limx3
x3 27
x4 81.
7. limx 1
2 sin(x + 1)
3x2 + 2x 1.
8. limn
n 1
n 4
4n+3
.
9.
f (x) =||x 2||(x 9)
x3 11x2 + 18x.
10.
f (x) =
2x+1, x ( ; 2],
2x2 x + 4, x ( 2; 2],
6x1, x (2; +) .
11. f (x) =5x2 + 11x + 42
x2 3x 10.
12. f (x) =5 4x 4 6x
4 4x + 3 6x.
. 66 473 2012/2013 .
-
MA , 1, 045
1. a) , ,
{xn} =
3 4n 8
5 4n a =
3
5,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
3n3 3n2 + 8n + 1
8n + 8.
3. limn
4n
3 + 8n2 9n + 2 6n + 8 cos
4n 5 3n
2 8n + 9.
4. limn
2 5n + 5 4n + 2
3 5n + 3 4n + 4.
5. limn
4 4n + 4n2
4n 41 8n + 5n2
5n 8.
6. limx2
3x2 + 5x 22
x5 32.
7. limx 1
6 tg(x + 1)
7x2 3x + 4.
8. limx1
x3x+2x1 .
9. f (x) =x + 4
||x2 + 6x + 8||.
10.
f (x) =
1x+4, x ( ; 4),
4x2 x + 3, x [ 4; 1),
8x+2, x ( 1; +) .
11. f (x) =5x3
x2 9.
12. f (x) =3 3x + 2 5x
2 3x + 3 5x.
. 67 473 2012/2013 .
-
MA , 1, 046
1. a) , ,
{xn} =
11 4n 7
3 4n 3
a =11
3,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
7n8 + 2n7 1
5n18 + 2n8 + 8n 6.
3. limn
n 8
2n
2 n + 9n
4 + 412
n
4 + 5n2 + 914 2n + 6
2 n 8
2.
4. limn
4n2 + 4 5n
2n2 2 5n.
5. limn
6 9n + 2n2
2n 51 + 5n + 5n2
5n + 2.
6. limx1
3x2 5x + 2
4x2 + 3x 7.
7. limx1
x2 + 8x 9
9 sin(x 1).
8. limx1
x3x7x1 .
9. f (x) =x 3
||x2 2x 3||.
10.
f (x) =
8x+3, x ( ; 2),
2x2 + 2x + 4, x [ 2; 3),
8x2, x [3; +) .
11. f (x) =5x2 35x + 60
x2 6x + 8.
12. f (x) =2 3x + 5 4x
4 3x 3 4x.
. 68 473 2012/2013 .
-
MA , 1, 047
1. a) , ,
{xn} =
8 5n + 11
5 5n + 8
a = 8
5,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
5n2 + 9n + 7
4n3 + 9n2 5.
3. limn
n
2 4 2n + 9
3
n
2 + 812 3n + 6
3.
4. limn
4 3n + 2 2n
3 2n 3 3n.
5. limn
16n
2 + n 2 4n .
6. limx 1
x3 + 1
2x2 + x 1.
7. limx1
2 sin(x 1)
x2 6x + 5.
8. limn
5n 1
5n 9
5n5
.
9.
f (x) =x2 5x 36
||x 9||(x2 + x 12).
10.
f (x) =
5x+3, x ( ; 3),
2x2 + 7x 4, x [ 3; 3),
9x+6, x [3; +) .
11. f (x) =x2 + 3x
x3 + 7x2 + 12x.
. 69 473 2012/2013 .
-
12. f (x) =4 2x 4 3x
5 2x + 5 3x.
. 70 473 2012/2013 .
-
MA , 1, 048
1. a) , ,
{xn} =
6 4n + 9
9 4n + 5 a =
2
3,
.
) , =1
1000 N,
n > N ||xn a|| < .
2. limn
5n12 + 5n + 1
4n20n + 3.
3. limn
3n
3 9n2 + 2n 3 n
3 2n2 + n 3
2n + 8
3n
2 + 7n + 9n
4 + 112
.
4. limn
5 3n + 4 6n 4
2 6n 2 3n + 2.
5. limn
9 3n 6n2
5n + 68 9n 6n2
5n 4.
6. limx3
x4 81
x3 27.
7. limx
(x + 7)3
7x + 3tg2
7
6x 7.
8. limn
4n + 5
4n + 2
7n6
.
9.
f (x) =||x + 3||(x + 5)
x3 + 8x2 + 15x.
10.
f (x) =
7x3, x ( ; 2),
x2 + 6x 1, x (2; 4),
7x3, x [4; +) .
11. f (x) =x3 4x2 10x 5
x2 + 2x + 1.
. 71 473 2012/2013 .
-
12. f (x) =5 3x 4 5x
4 3x 3 5x.
. 72 473 2012/2013 .
-
MA , 1, 049
1. a) , ,
{xn} =
2 3n + 3
9 3n 6 a =
2
9,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
3n14n8 + 2
2n16 + 4n 6.
3. limn
n7n + 4 cos n 4 6n 4
6n 5
3 .
4. limn
5 4n 4 5n + 5
2 4n 5 5n + 4.
5. limn
4 + n + 8n2
8n + 16 + 5n + 6n2
6n + 5.
6. limx 3
x3 + 27
x4 81.
7. limx
9x4 + 5x2 8x 2
2x + 6sin3
3
2x 6.
8. limx
6x9 + 1
6x9 4
7x9+7.
9.
f (x) =x2 + 2x 35
||x + 7||(x2 2x 15).
10.
f (x) =
1x+2, x ( ; 3),
2x2 5x + 2, x [ 3; 2),
4x+3, x ( 2; +) .
11. f (x) =3x3 + 10x2 x 6
x2 4x + 3.
. 73 473 2012/2013 .
-
12. f (x) =2 3x 3 5x
2 3x 3 5x.
. 74 473 2012/2013 .
-
MA , 1, 050
1. a) , ,
{xn} =
7n 1
5n + 3
a = 7
5,
.
) , =13
10000 N,
n > N ||xn a|| < .
2. limn
4n7 6n6 1
7n12 + 2.
3. limn
2n
2 + 6n cos(6n 7) 6n 6
9n 8
3 .
4. limn
4 5n + 5 6n 3
5 6n + 5 5n 4.
5. limn
6 + 5n + 5n2
8n 23 7n + 5n2
8n + 2.
6. limx2
2x2 + x + 6
x4 16.
7. limx1
x2 8x + 7
6 tg(x 1).
8. limx
3x + 2
3x 1
5x4.
9.
f (x) =||x||(x + 3)
x3 5x2 24x.
10.
f (x) =
8x+6, x ( ; 6),
x2 6x 4, x ( 6; 3],
5x+4, x ( 3; +) .
11. f (x) =2x2 2x 12
x3 2x2 8x.
. 75 473 2012/2013 .
-
12. f (x) =4 3x + 4 5x
2 3x 3 5x.
. 76 473 2012/2013 .
-
MA , 1, 051
1. a) , ,
{xn} =
2 4n 6
7 4n + 10 a =
2
7,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
1
8n3 + 4.
3. limn
n
4 + 814n + 7
4
3n + 7
2n
2 1 2n 9
2 .
4. limn
3n5 4 3n n + 1
3n4 2 6n 5n 4.
5. limn
4n 16n
2 + 4n 3 .
6. limx 1
3x2 + 5x + 8
3x2 + 2x 1.
7. limx
x5 + 3x3 + 8x 7
9x + 3sin4
2
3x 7.
8. limn
(n 7)ln(2n 7) ln(2n + 4) .
9.
f (x) =x2 + 13x + 42
||x + 6||(x2 + 12x + 35).
10.
f (x) =
5x8, x ( ; 5),
3x2 + 2x + 4, x [ 5; 1),
6x+7, x ( 1; +) .
11. f (x) =x3 + 2x2 + 5x 5
x2 4.
12. f (x) =5 4x 5 5x
5 4x + 2 5x.
. 77 473 2012/2013 .
-
MA , 1, 052
1. a) , ,
{xn} =
9n + 11
3n 4
a = 3,
.
) , =11
10000 N,
n > N ||xn a|| < .
2. limn
6n4 + 6n3 + 6n + 5
7n5 + 5n2.
3. limn
6n + 8
2 3n
2 + 82
6n
6 + 9.
4. limn
3 6n 5n4 + 3
5 6n + 3n4 2.
5. limn
3 + 6n + 6n2
6n + 66 + 4n + 5n2
5n + 4.
6. limx 2
x4 16
x5 + 32.
7. limx
6x5 3x4 + 2x + 4
x + 7sin4
1
3x 4.
8. limn
7n 7
7n + 2
6 7n+9
.
9. f (x) =||x + 8||(x 7)
x3 + x2 56x.
10.
f (x) =
2x+3, x ( ; 4],
5x2 + 3x 5, x ( 4; 6),
3x5, x (6; +) .
11. f (x) =3x3 7x2 3x + 10
x2 x 2.
12. f (x) =4 2x + 2 3x
3 2x + 2 3x.
. 78 473 2012/2013 .
-
MA , 1, 053
1. a) , ,
{xn} =
4n + 2
9n + 5
a = 4
9,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
7n16 7n10 + 2n + 3
4n6 5.
3. limn
7n + 2
2 4n
2 62
2n
2 43 .
4. limn
3 7n + 3n4 3
4 7n + 4n4 4.
5. limn
2 5n + 6n2
6n 27 + 2n 8n2
2n + 8.
6. limx2
3x2 x 10
6x2 + 5x + 14.
7. limx
3x5 2x3 9x 1
2x 5sin4
1
3x 9.
8. limn
(2n 1)ln(3n 4) ln(3n + 7) .
9. f (x) =x + 3
||x2 + 4x + 3||.
10.
f (x) =
2x+5, x ( ; 6),
x2 6x + 2, x ( 6; 5],
5x+5, x ( 5; +) .
11. f (x) =2x3 4x2 x + 2
x2 3x + 2.
12. f (x) =2 4x + 2 5x
3 4x 4 5x.
. 79 473 2012/2013 .
-
MA , 1, 054
1. a) , ,
{xn} =
5n + 4
5n + 4
a = 1,
.
) , =3
100000 N,
n > N ||xn a|| < .
2. limn
n + 5
16n2 4n 1 + 9n2 + 4n + 4.
3. limn
7n
3 4n2 4n 59n 8 cos
7( 5n + 9)8n 4
2 .
4. limn
5n6 + 3 4n n + 4
3n3 + 4 5n 5n 3.
5. limn
1 n
2 4n + 1 n2 4n 1 (4n + 4).
6. limx4
2x2 + 6x + 8
2x2 + 4x + 16.
7. limx
4x3
x + 1sin2
1
5x + 7.
8. limn
4n 5
4n + 1
4n4
.
9.
f (x) =||x||(x + 3)
x3 + 10x2 + 21x.
10.
f (x) =
4x+1, x ( ; 2),
5x2 + 2x 1, x [ 2; 4],
5x3, x (4; +) .
11. f (x) =2x3 + 3x2 + 3x 4
x2 1.
. 80 473 2012/2013 .
-
12. f (x) =4 5x 4 6x
4 5x + 2 6x.
. 81 473 2012/2013 .
-
MA , 1, 055
1. a) , ,
{xn} =
5 3n + 3
9 3n 7
a =5
9,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
16n2 + 5n 3 + n2 + 4n + 2
n + 1.
3. limn
7n
3 9n2 8n 33n
2 + 8n + 53n 6 cos n + 8
.
4. limn
5 3n 4 6n + 4
3 3n + 4 6n 2.
5. limn
4 + 6n + 9n2
9n + 65 4n + 7n2
7n 4.
6. limx 2
3x2 + 2x 8
x2 x + 2.
7. limx
6x5 + 5x + 8
5x 3tg4
2
3x 7.
8. limx0
3 + x
3 + 4x
1tg5x .
9.
f (x) =||x 8||(x + 6)
x3 2x2 48x.
10.
f (x) =
6x2, x ( ; 1),
7x2 x 6, x [ 1; 6],
1x6, x (6; +) .
11. f (x) =x3 6x2 4x + 8
x2 + x 2.
. 82 473 2012/2013 .
-
12. f (x) =2 4x 2 5x
5 4x 4 5x.
. 83 473 2012/2013 .
-
MA , 1, 056
1. a) , ,
{xn} =
2n 8
5n + 3 a =
2
5,
.
) , =3
1000 N,
n > N ||xn a|| < .
2. limn
2n 5 cos68n 7
5n 6 sin4n + 4.
3. limn
2n + 3
6
3n
2 22n
2 + 42 .
4. limn
4 6n + 4n3 4
3 6n 3n3 3.
5. limn
2 4n
2 4 4n2 + 4 (4n + 2).
6. limx 2
5x2 2x 24
x5 + 32.
7. limx
7x4 4x 1
5x 7sin3
1
2x 6.
8. limx1
x5x+21 x .
9.
f (x) =||x 4||(x + 7)
x3 + 3x2 28x.
10.
f (x) =
3x+5, x ( ; 6],
5x2 + 3x 4, x ( 6; 4],
4x3, x (4; +) .
11. f (x) =5x3 3x2 3x + 5
x2 1.
. 84 473 2012/2013 .
-
12. f (x) =2 4x 3 5x
2 4x + 3 5x.
. 85 473 2012/2013 .
-
MA , 1, 057
1. a) , ,
{xn} =
3 5n + 5
9 5n + 4
a =1
3,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
8n20 + 4n + 6
9n8 + 3n5 4n 8.
3. limn
9n
2 + 45n
2 + 99n
2 + 87n
2 + 13
.
4. limn
2 4n + 4 5n
2 5n 2 4n.
5. limn
4 + 5n + 3n2
3n + 57 n + 8n2
8n 1.
6. limx2
x2 3x + 10
x2 3x + 10.
7. limx
( x 7)3
4x + 3sin2
5
6x 6.
8. limx
4x + 1
4x 4
3x4.
9. f (x) =||x + 1||(x 2)
x3 x2 2x.
10.
f (x) =
3x9, x ( ; 6),
7x2 + 5x 9, x ( 6; 2),
9x+3, x ( 2; +) .
11. f (x) =3x3 + 5x2 x 5
x2 + 2x + 1.
12. f (x) =4 4x + 4 5x
3 4x + 2 5x.
. 86 473 2012/2013 .
-
MA , 1, 058
1. a) , ,
{xn} =
5 3n 2
4 3n 5
a =5
4,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
3n2 + 9
5.
3. limn
3n
2 + 65n
2 92
9n
6 9.
4. limn
5 6n 2n3 4
2 6n 4n3 + 3.
5. limn
6 + 6n + 7n2
7n + 66 + 7n + 2n2
2n + 7.
6. limx0
5x + 25 5
2x + 49 7.
7. limx 1
7 sin(x + 1)
9x2 + 7x 2.
8. limx
x6 + 7
x6 1
4x67.
9.
f (x) =x2 3x 4
||x + 1||(x2 6x + 8).
10.
f (x) =
5x9, x ( ; 8),
9x2 + 4x 6, x ( 8; 1),
6x+7, x ( 1; +) .
11. f (x) =x3 4x2 2
x2 4.
. 87 473 2012/2013 .
-
12. f (x) =2 3x 5 6x
2 3x + 2 6x.
. 88 473 2012/2013 .
-
MA , 1, 059
1. a) , ,
{xn} =
7 2n 7
6 2n + 4 a =
7
6,
.
) , =1
1000000000 N,
n > N ||xn a|| < .
2. limn
2n8 n7 + 4n5 + 8n3 + 6n2 4n 8
6n8 4n7 8n6 6n5 + 4n3 + 3.
3. limn
8n
2 1 n
2 6 6n
2 65n
3 + 92 .
4. limn
2 3n 5 6n + 3
4 6n 3 3n 3.
5. limn
2 8n + 8n2
8n 86 + 5n + n2
n + 5.
6. limx 2
6x2 + 3x 18
2x2 + 3x + 14.
7. limx
(x 1)3
3x 9tg2
9
4x + 8.
8. limn
3n + 9
3n + 4
3n3
.
9.
f (x) =x2 + 15x + 56
||x + 7||(x2 + 12x + 32).
10.
f (x) =
2x+2, x ( ; 3),
x2 3x + 2, x [ 3; 1],
4x+2, x ( 1; +) .
11. f (x) =2x3 2x2 + x 2
x2 2x + 1.
. 89 473 2012/2013 .
-
12. f (x) =4 4x + 3 5x
2 4x + 3 5x.
. 90 473 2012/2013 .
-
MA , 1, 060
1. a) , ,
{xn} =
5 3n + 7
8 3n + 5 a =
5
8,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
9n 6 sin58n + 3
5n + 4 sin5n + 5.
3. limn
2n + 5
42n 8
2
3n + 3
4n
2 + 612
.
4. limn
2 7n 3 6n
4 6n 5 7n.
5. limn
2n 4n
2 + 5n 2 .
6. limx 1
6x2 2x 8
x5 + 1.
7. limx0
4x2arctg2x
2 tg( 9x + 2x
2)
tg(3x2) + 4 sin( 7x).
8. limn
3n 6
3n + 1
3n+9
.
9. f (x) =x + 4
||x2 + 6x + 8||.
10.
f (x) =
3x+2, x ( ; 3],
x2 4x + 6, x ( 3; 5],
3x4, x (5; +) .
11. f (x) =4x2 16x
x2 + 5x + 4.
12. f (x) =3 2x 2 3x
3 2x 2 3x.
. 91 473 2012/2013 .
-
MA , 1, 061
1. a) , ,
{xn} =
2 3n + 9
6 3n + 2 a =
1
3,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
7n5 + n4 + 5n3 8n2 + 1
6n 4.
3. limn
n + 2
4 2n 1
2
n5 2n5
2n 33n 8
4.
4. limn
2n2 5 6n 4n 4
4n7 + 5 5n + 3.
5. limn
8 7n + 5n2
8n 21 3n + 8n2
7n 6.
6. limx6
x 6
4x + 105 9.
7. limx
( 4x 5)3
x + 6tg2
1
8x + 3.
8. limx
5x4 + 9
5x4 3
3x47.
9.
f (x) =x2 7x 18
||x 9||(x2 + 10x + 16).
10.
f (x) =
3x4, x ( ; 3),
x2 + 2x + 6, x [3; 4],
2x5, x (4; +) .
11. f (x) =3x3 9x2 2x + 5
x2 + 4x + 3.
. 92 473 2012/2013 .
-
12. f (x) =2 2x 3 3x
4 2x + 5 3x.
. 93 473 2012/2013 .
-
MA , 1, 062
1. a) , ,
{xn} =
11 5n 4
5 5n + 5
a =11
5,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
n 7 cos53n + 3
5n 6 sin6n + 8.
3. limn
6n
2 + 3n + 28n 3 sin
2( 5n 4)
n8n + 52 .
4. limn
2 3n 4 2n
3 2n 5 3n.
5. limn
2 + 5n 4n2
3n 45 + 6n 4n2
3n 8.
6. limx 1
2x2 + x + 3
6x2 + 4x 2.
7. limx
(5x + 7)2
x + 2tg
1
3x 5.
8. limx1
x5x+2x1 .
9. f (x) =x 2
||x2 3x + 2||.
10.
f (x) =
1x1, x ( ; 2),
2x2 3x 3, x [2; 3],
6x2, x (3; +) .
11. f (x) =2x3 + 9x2 10x + 2
x2 4x + 4.
12. f (x) =4 2x 3 5x
4 2x 2 5x.
. 94 473 2012/2013 .
-
MA , 1, 063
1. a) , ,
{xn} =
5 4n
6 4n + 8
a = 5
6,
.
) , =1
1000000000 N,
n > N ||xn a|| < .
2. limn
6n5 + 5n3 + 2n
4n4 + 6n3 + 8n2 6n + 2.
3. limn
2n + 8
n
2 + 3 n + 6
2n + 4
2
4n
2 + 7n + 4n
2 532
.
4. limn
2 5n + 2 3n
5 5n 2 3n.
5. limn
1 + n + 7n2
7n + 17 3n + n2
n 3.
6. limx2
4x2 + 4x + 8
x2 4.
7. limx
2x2 7x + 6
x 4tg
5
8x + 5.
8. limn
(3n 6)ln(8n 1) ln(8n 8) .
9. f (x) =||x||(x + 1)
x3 3x2 4x.
10.
f (x) =
8x+1, x ( ; 2),
x2 + 3x + 2, x ( 2; 1],
2x+4, x ( 1; +) .
11. f (x) =3x3 8x2 2x + 7
x2 3x + 2.
12. f (x) =3 2x 4 3x
3 2x + 2 3x.
. 95 473 2012/2013 .
-
MA , 1, 064
1. a) , ,
{xn} =
8 4n + 8
10 4n 8
a =4
5,
.
) , =1
1000000000 N,
n > N ||xn a|| < .
2. limn
4n 2
5n + 3 + 9n2 2n + 3.
3. limn
n 6n2 2n + 6
4n + 2 cos n 7
6n + 3
2 .
4. limn
3 6n 3 5n
3 6n + 3 5n.
5. limn
7 + 6n + 3n2
3n + 63 n + 9n2
9n 1.
6. limx4
x4 256
x2 2x + 24.
7. limx2
4x2 + 5x + 6
7 tg(x 2).
8. limx1
x4x51 x .
9.
f (x) =||x 5||(x 9)
x3 14x2 + 45x.
10.
f (x) =
4x3, x ( ; 2),
x2 + 2x + 4, x [2; 4],
4x5, x (4; +) .
11. f (x) =3x3 x2 2
x2 4.
. 96 473 2012/2013 .
-
12. f (x) =4 3x 3 5x
3 3x 3 5x.
. 97 473 2012/2013 .
-
MA , 1, 065
1. a) , ,
{xn} =
6 5n 4
7 5n 5 a =
6
7,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
6n18 + 4n9 9
5n9 + 9n7 + 1.
3. limn
2n
3 + n2 8n
2 + 112
n7 + 3n7 n 3 n
2 + 7.
4. limn
5n3 + 2 6n 3n 4
3n5 + 3 3n + 3.
5. limn
3 + 4n + 5n2
5n + 46 + 7n + 5n2
5n + 7.
6. limx3
x2 3x + 18
x2 + 5x 24.
7. limx
(3x + 6)2
8x 4tg
3
8x + 5.
8. limn
5n + 8
5n + 3
6n8
.
9.
f (x) =x2 13x + 40
||x 8||(x2 + 3x 40).
10.
f (x) =
2x+3, x ( ; 6],
6x2 2x 7, x ( 6; 6],
6x6, x (6; +) .
11. f (x) =2x2 6x
x3 6x2 + 8x.
. 98 473 2012/2013 .
-
12. f (x) =3 2x 5 3x
4 2x 4 3x.
. 99 473 2012/2013 .
-
MA , 1, 066
1. a) , ,
{xn} =
2n + 3
6n a =
1
3,
.
) , =13
10000 N,
n > N ||xn a|| < .
2. limn
7n7 + 8n6 + 7n5 4n4 n3 + 4n 4
6n4 3n3 5n2 + n 4.
3. limn
n 2n + 4 3n 3
2
n7 + n7 n 23n 8
.
4. limn
3n6 2 2n
2n6 + 3 2n.
5. limn
4 8n + 5n2
4n 76 + 3n + 5n2
4n 3.
6. limx0
9x
8x + 49 7.
7. limx
8x3 4x2
9x 5sin2
5
3x 4.
8. limn
5n 1
5n + 3
6 5n+9
.
9.
f (x) =x2 12x + 32
||x 4||(x2 4x 32).
10.
f (x) =
6x+3, x ( ; 2],
2x2 x 4, x ( 2; 1),
1x+2, x [ 1; +) .
11. f (x) =2x2 + 7x 5
x2 3x + 2.
. 100 473 2012/2013 .
-
12. f (x) =2 5x 4 6x
2 5x 2 6x.
. 101 473 2012/2013 .
-
MA , 1, 067
1. a) , ,
{xn} =
10 3n
8 3n + 6 a =
5
4,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
7n6 + 5n3 8n2 + 7n 3
6n8 + 6n4 n3 + n2 + 2n + 2.
3. limn
3n + 7
2n
2 332
2n 2
n
2 + 1n
4 312
.
4. limn
3 6n 2n6 + 3
4n6 + 4 6n + 4.
5. limn
n n
2 + 2n 1 .
6. limx 4
2x2 + 6x 8
x2 16.
7. limx 2
3x2 8x 4
7 sin(x + 2).
8. limx
7x8 4
7x8 6
x85.
9. f (x) =||x 1||
x2 + 3x 4.
10.
f (x) =
2x+3, x ( ; 6),
7x2 9x 7, x ( 6; 1),
9x+2, x ( 1; +) .
11. f (x) =2x3 7x2 + 7x + 3
x2 4x + 4.
12. f (x) =3 2x 3 6x
2 2x 3 6x.
. 102 473 2012/2013 .
-
MA , 1, 068
1. a) , ,
{xn} =
7n + 10
9n + 10
a = 7
9,
.
) , =1
10 N,
n > N ||xn a|| < .
2. limn
9n2 + 5
3n5 + 5n3 + n2 + 4n.
3. limn
n4n2 + 7n 6
4n
2 + 2n + 47n 3 sin
7( 9n 3).
4. limn
4n5 2 2n 2
3n4 + 3 3n n + 4.
5. limn
3 7n 8n2
2n 98 3n 8n2
2n 2.
6. limx4
2x2 2x 24
x3 64.
7. limx 2
2x2 + 5x + 2
5 tg(x + 2).
8. limx1
x9x2x1 .
9.
f (x) =x2 11x + 18
||x 9||(x2 + 5x 14).
10.
f (x) =
8x5, x ( ; 5),
6x2 + 6x 5, x (5; 7],
7x+2, x (7; +) .
11. f (x) =5x2 + 27x 34
x2 5x + 6.
. 103 473 2012/2013 .
-
12. f (x) =4 3x 5 4x
5 3x + 2 4x.
. 104 473 2012/2013 .
-
MA , 1, 069
1. a) , ,
{xn} =
8n 4
11n 2
a = 8
11,
.
) , =3
10000 N,
n > N ||xn a|| < .
2. limn
8n4 + 5n3 + 9
7n 8.
3. limn
n2n 2n
2 + 632
n6 26
2n 12 3n
2 9
.
4. limn
2 7n 2n7 2
2n7 + 2 7n + 4.
5. limn
2 n + 7n2
7n 13 + 6n + 4n2
4n + 6.
6. limx3
x4 81
x3 27.
7. limx
(x 6)2
3x + 4sin
3
x 6.
8. limx
2x + 3
2x 2
2x3.
9. f (x) =||x + 1||
x2 + 6x + 5.
10.
f (x) =
8x7, x ( ; 1],
x2 + 3x + 3, x ( 1; 8],
6x2, x (8; +) .
11. f (x) =3x2 + 14x + 17
x2 3x 4.
12. f (x) =2 3x + 3 4x
2 3x 2 4x.
. 105 473 2012/2013 .
-
MA , 1, 070
1. a) , ,
{xn} =
8n + 6
2n 2
a = 4,
.
) , =3
10000 N,
n > N ||xn a|| < .
2. limn
7n4 + 2n3 8
3n14 + 2n9 + 3.
3. limn
2n
2 73
4n 1
27n
2 + 72 .
4. limn
2n3 + 3 + 3 6n
4 6n 2n3 + 4.
5. limn
2 5n + 8n2
7n 33 + 2n 5n2
6n 6.
6. limx 2
5x2 + 6x 8
x3 + 8.
7. limx0
5x2arctg4x
+ 2 tg( 5x)
2 sin( 2x) + tg( 5x2).
8. limn
9n 8
9n + 1
8n+3
.
9. f (x) =x 3
||x2 + 2x 15||.
10.
f (x) =
6x3, x ( ; 3),
4x2 x 4, x [3; 7],
3x7, x (7; +) .
11. f (x) =3x2 + 6x + 9
x2 x 6.
. 106 473 2012/2013 .
-
12. f (x) =3 4x 2 6x
5 4x + 2 6x.
. 107 473 2012/2013 .
-
MA , 1, 071
1. a) , ,
{xn} =
7n + 5
n + 8
a = 7,
.
) , =11
100000 N,
n > N ||xn a|| < .
2. limn
7n20 + 1
8n20 7.
3. limn
6n
2 + 5n 5 5n + 7 sin
4( 2n + 4)n8n
2 2n + 4.
4. limn
3 4n 3 2n 2
2 4n + 3 2n + 2.
5. limn
1 + 4n + 7n2
7n + 46 + 2n + 8n2
8n + 2.
6. limx 2
2x2 + 2x + 12
x2 + 3x + 2.
7. limx
9x4 + x 6
8x + 4sin3
1
2x 2.
8. limx
7x8 9
7x8 4
4x81.
9.
f (x) =x2 + 9x + 18
||x + 6||(x2 + 10x + 21).
10.
f (x) =
9x+3, x ( ; 2),
x2 + 5x + 5, x [ 2; 1],
1x+2, x ( 1; +) .
11. f (x) =4x2 + 5x 16
x2 + 2x 3.
. 108 473 2012/2013 .
-
12. f (x) =5 2x 4 3x
5 2x 3 3x.
. 109 473 2012/2013 .
-
MA , 1, 072
1. a) , ,
{xn} =
9 3n 8
5 3n + 7 a =
9
5,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
3n2 3n 4
2.
3. limn
3n 3
2 3n
2 + 22
5n
2 + 33
.
4. limn
3n3 + 4 4n
2n6 3 3n + 1.
5. limn
1 + 3n + 6n2
6n + 36 6n + 6n2
6n 6.
6. limx4
x4 256
x2 6x + 8.
7. limx1
7 tg(x 1)
x2 + 6x 7.
8. limx
5x5 6
5x5 + 2
3x58.
9. f (x) =||x 3||(x + 1)
x3 2x2 3x.
10.
f (x) =
5x+2, x ( ; 7),
x2 8x 8, x ( 7; 6),
6x3, x (6; +) .
11. f (x) =3x2 10x 13
x2 3x 4.
12. f (x) =3 4x + 2 5x
2 4x + 4 5x.
. 110 473 2012/2013 .
-
MA , 1, 073
1. a) , ,
{xn} =
n + 8
4n + 1
a = 1
4,
.
) , =7
100 N,
n > N ||xn a|| < .
2. limn
2n7 2
5n14 + 5n9 6.
3. limn
5n
2 43
7n 6
25n
2 72 .
4. limn
3 7n 5 2n + 2
4 7n + 4 2n + 3.
5. limn
1 + 2n + 6n2
6n + 26 3n + 4n2
4n 3.
6. limx8
x 8
3x 8 4.
7. limx
(x + 1)2
5x + 3sin
7
2x 8.
8. limx
9x5 + 7
9x5 + 3
x52.
9. f (x) =||x + 2||
x2 + 6x + 8.
10.
f (x) =
5x+3, x ( ; 4],
2x2 + 3x + 5, x ( 4; 2],
6x+3, x ( 2; +) .
11. f (x) =3x2 3x + 36
x3 + 2x2 8x.
12. f (x) =5 3x + 2 6x
2 3x + 4 6x.
. 111 473 2012/2013 .
-
MA , 1, 074
1. a) , ,
{xn} =
8n + 4
n + 10 a = 8,
.
) , =17
1000 N,
n > N ||xn a|| < .
2. limn
9n + 6 sin3n 3
3n 5 cos5n + 4.
3. limn
5n 8 cos
3n + 5 8n
2 + 5n 3n2 7n 2
.
4. limn
3 6n 5 4n
4 6n + 2 4n.
5. limn
4 3n + 9n2
9n 32 + 4n + 4n2
4n + 4.
6. limx 4
x3 + 64
x2 16.
7. limx
5x4 4x + 6
7x + 1sin3
1
2x 2.
8. limn
(3n + 2)ln(9n 1) ln(9n 3) .
9.
f (x) =x2 6x 7
||x + 1||(x2 16x + 63).
10.
f (x) =
6x2, x ( ; 1),
3x2 + 5x + 4, x [ 1; 8],
7x+4, x (8; +) .
11. f (x) =x2 + 2x + 3
x2 4x + 3.
12. f (x) =3 3x + 4 4x
3 3x 2 4x.
. 112 473 2012/2013 .
-
MA , 1, 075
1. a) , ,
{xn} =
10n 3
11n + 11
a =10
11,
.
) , =7
100 N,
n > N ||xn a|| < .
2. limn
4n6 + 8
5n16 2n8 5.
3. limn
3n + 6
2 3n + 5
4
n + 9
2 n + 8
4 .
4. limn
4n3 5 6n 4n 5
2n4 + 4 3n 3n.
5. limn
7 6n 3n2
6n 34 8n + n2
4n 6.
6. limx2
4x2 + 2x + 12
6x2 4x 16.
7. limx
( 8x 2)2
9x + 5sin
1
2x + 1.
8. limx
3x3 6
3x3 + 2
9x33.
9.
f (x) =x2 + x 20
||x 4||(x2 3x 40).
10.
f (x) =
7x8, x ( ; 2),
x2 + 9x 4, x (2; 9),
4x8, x (9; +) .
11. f (x) =x3 5x2 x + 5
x2 3x 4.
. 113 473 2012/2013 .
-
12. f (x) =4 3x + 3 5x
5 3x + 3 5x.
. 114 473 2012/2013 .
-
MA , 1, 076
1. a) , ,
{xn} =
7 4n + 10
3 4n 6
a = 7
3,
.
) , =1
100000 N,
n > N ||xn a|| < .
2. limn
7n2 + 1
2.
3. limn
8n
2 7 n
2 82
3n
6 + 4.
4. limn
3n5 3 7n 3n 4
4n7 + 3 3n.
5. limn
16n
2 3n + 1 4n .
6. limx 2
4x2 + 5x 6
x5 + 32.
7. limx
6x5 + 2x + 8
7x + 3sin4
3
2x 3.
8. limx1
xx+91 x .
9.
f (x) =x2 6x + 8
||x 4||(x2 7x + 10).
10.
f (x) =
3x+2, x ( ; 3],
2x2 3x + 6, x ( 3; 1),
7x+2, x ( 1; +) .
11. f (x) =4x2 + 30x + 50
x2 + 7x + 10.
12. f (x) =4 2x 3 5x
2 2x + 3 5x.
. 115 473 2012/2013 .
-
MA , 1, 077
1. a) , ,
{xn} =
n + 3
11n + 2
a =1
11,
.
) , =1
100 N,
n > N ||xn a|| < .
2. limn
4n2 + 2n + 3 + 4n2 + 5n 3
3n 3.
3. limn
n + 7
44n + 9
2 3n 5
2
n
4 + n3 + 6 2n 4
3 .
4. limn
2n3 3 7n n 2
3n6 2 3n 4n 4.
5. limn
25n
2 + 4n + 5 5n .
6. limx 2
x5 + 32
x4 16.
7. limx
5x4 8x + 7
3x + 8tg3
3
2x 4.
8. limn
5n 9
5n + 8
3 5n7
.
9. f (x) =||x 1||
x2 3x + 2.
10.
f (x) =
4x+5, x ( ; 6),
5x2 + 4x 1, x ( 6; 2],
4x1, x (2; +) .
11. f (x) =3x2 + 6x + 24
x3 9x2 + 20x.
12. f (x) =3 4x + 4 5x
4 4x 3 5x.
. 116 473 2012/2013 .
-
MA , 1, 078
1. a) , ,
{xn} =
6 3n 5
5 3n 3 a =
6
5,
.
) , =1
1000000000 N,
n > N ||xn a|| < .
2. limn
n3 5n2 + 8n 3
7n3 8n2 7n.
3. limn
4n
3 + 62
8n
2 + 54n
2 42.
4. limn
2 6n + 3n5 + 4
2n5 + 4 6n + 2.
5. limn
5 + 5n 8n2
2n + 37 + 7n 5n2
n + 4.
6. limx 1
2x2 + 2x + 4
5x2 4x 9.
7. limx
9x3
4x + 8sin2
5
6x 6.
8. limx
4x 2
4x 4
3x+5.
9. f (x) =||x + 1||
x2 3x 4.
10.
f (x) =
3x+3, x ( ; 4),
2x2 + 6x 5, x ( 4; 3),
5x+4, x [ 3; +) .
11. f (x) =x3 x2 + 7x 2
x2 x 2.
12. f (x) =4 3x + 4 4x
2 3x 5 4x.
. 117 473 2012/2013 .
-
MA , 1, 079
1. a) , ,
{xn} =
11 3n + 6
3 3n 7
a =11
3,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
6n10 2n3 8
7n20 + 3n9 6n3 2.
3. limn
9n
2 + 5 5n
2 + 63n
2 22n
3 42 .
4. limn
2 5n + 4n4 + 2
4 5n 2n4 + 4.
5. limn
7 + n + 6n2
n + 62 + 7n 8n2
8n + 8.
6. limx2
3x2 + 5x 22
x2 x + 6.
7. limx1
5 sin(x 1)
6x2 7x + 1.
8. limn
3n + 1
3n 6
3n4
.
9.
f (x) =x2 3x 40
||x + 5||(x2 5x 24).
10.
f (x) =
7x+4, x ( ; 4),
4x2 + 7x 1, x ( 4; 1),
4x+2, x ( 1; +) .
11. f (x) =4x3 x2 1
x2 + 3x 10.
. 118 473 2012/2013 .
-
12. f (x) =3 3x + 3 4x
2 3x + 5 4x.
. 119 473 2012/2013 .
-
MA , 1, 080
1. a) , ,
{xn} =
3n + 3
n + 7 a = 3,
.
) , =7
10 N,
n > N ||xn a|| < .
2. limn
9n2 8
3n6 n5 + 6n3 + 8n + 8.
3. limn
3n
2 5 2n 3
4
2n 5
4n
2 3n
2 + 332
.
4. limn
2 6n + 4n6 + 3
4 6n 2n6 3.
5. limn
4
4n2 n + 1 2n + 1.
6. limx0
6x + 25 5
5x + 16 4.
7. limx0
9x2cos6x
+ 6 sin( 4x)
tg(5x2) 2 sin( 4x).
8. limx1
x3x8x1 .
9. f (x) =x + 1
||x2 2x 3||.
10.
f (x) =
9x+2, x ( ; 3),
2x2 4x 3, x [ 3; 2],
3x+3, x ( 2; +) .
11. f (x) =4x3
x2 9.
12. f (x) =4 3x + 4 4x
4 3x + 3 4x.
. 120 473 2012/2013 .
-
MA , 1, 081
1. a) , ,
{xn} =
8 3n + 2
9 3n + 7 a =
8
9,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
7n + 3 sin52n 7
3n + 6 cos3n 2.
3. limn
2n 8
2n 6
n + 4
3
3n + 2
22n + 6
4
.
4. limn
2 7n + 2 4n 4
3 7n 2 4n 5.
5. limn
3
2n + 1 4n2 3n 4.
6. limx4
x3 64
x2 + x 20.
7. limx
5x3
2x + 9tg2
7
6x + 2.
8. limn
6n 1
6n 4
3n+3
.
9.
f (x) =x2 + x 12
||x + 4||(x2 x 6).
10.
f (x) =
6x+2, x ( ; 3),
6x2 + 3x 4, x [ 3; 6),
3x5, x [6; +) .
11. f (x) =5x2 32x 48
x2 + 6x + 8.
. 121 473 2012/2013 .
-
12. f (x) =5 3x 2 6x
2 3x + 5 6x.
. 122 473 2012/2013 .
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MA , 1, 082
1. a) , ,
{xn} =
n + 7
5n + 3
a = 1
5,
.
) , =3
10000 N,
n > N ||xn a|| < .
2. limn
3n16 n + 9
5n20 3n8 1.
3. limn
n + 5
2n + 9
32n 3
n 7
2n
2 + 5n 23n
3 5n + 3.
4. limn
3n6 4 + 5 4n
2 4n 4n6 3.
5. limn
3 n
2 + 3 n2 4 (n 5).
6. limx 1
x2 1
x5 + 1.
7. limx1
x2 2x + 1
5 sin(x 1).
8. limx1
x4x+1x1 .
9.
f (x) =x2 + 15x + 54
||x + 9||(x2 + 10x + 24).
10.
f (x) =
8x+7, x ( ; 7),
2x2 8x + 7, x ( 7; 3],
1x2, x (3; +) .
11. f (x) =x2 x + 18
x2 2x 15.
. 123 473 2012/2013 .
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12. f (x) =5 2x + 3 5x
4 2x 3 5x.
. 124 473 2012/2013 .
-
MA , 1, 083
1. a) , ,
{xn} =
9 4n + 4
8 4n + 10
a =9
8,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
9n2 3n 2 4n2 2n + 1
3n 5.
3. limn
3n + 2
22n + 5
2n
4 + 112
n 8
3n
2 2n + 8 2n + 1
2.
4. limn
2n4 + 3 5n + 2n
3n5 + 2 3n 2n 2.
5. limn
3 + 9n + 5n2
3n + 43 3n n2
8n 5.
6. limx 2
4x2 2x 20
2x2 + 6x + 20.
7. limx
2x5 + 4x + 5
9x + 7sin4
3
x 7.
8. limn
7n + 2
7n 7
n+2
.
9.
f (x) =||x 2||(x 5)
x3 7x2 + 10x.
10.
f (x) =
9x2, x ( ; 1],
5x2 2x 4, x ( 1; 4],
2x4, x (4; +) .
11. f (x) =3x3 + 2x2 4
x2 25.
. 125 473 2012/2013 .
-
12. f (x) =2 3x + 5 6x
3 3x 2 6x.
. 126 473 2012/2013 .
-
MA , 1, 084
1. a) , ,
{xn} =
10 3n + 9
3 3n + 11
a =10
3,
.
) , =1
10000 N,
n > N ||xn a|| < .
2. limn
8n14 5
5n18 + n 7.
3. limn
n
2 + 112 n 6
2
n10 + 15 2n 6
.
4. limn
2 7n 3 6n 2
4 7n + 2 6n + 4.
5. limn
1
25n2 4n + 3 25n2 5n + 4.
6. limx 8
7x + 120 8
4x + 113 9.
7. limx9
x2 8x 9
8 sin(x 9).
8. limn
3n + 7
3n + 1
3n1
.
9. f (x) =||x + 1||
x2 2x 3.
10.
f (x) =
4x+2, x ( ; 3],
2x2 + 4x 2, x ( 3; 1),
4x+2, x [ 1; +) .
11. f (x) =3x2 19x 28
x2 + 6x + 8.
12. f (x) =2 3x 2 4x
4 3x 5 4x.
. 127 473 2012/2013 .
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MA , 1, 085
1. a) , ,
{xn} =
7 3n 4
3 3n 8 a =
7
3,
.
) , =1
1000000 N,
n > N ||xn a|| < .
2. limn
5n4 8n2 8
8n4 4n.
3. limn
4n + 7
n
2 + 4n 43n
4 8n3 4n2 9
3n + 2
2n
2 + 2n 3n
2 632
.
4. limn
3 6n + 2 5n 3
2 6n 4 5n 4.
5. limn
8 + 7n 3n2
6n 49 6n + 9n2
5n 2.
6. limx 2
3x2 4x 20
x2 5x 6.
7. limx1
3x2 + 4x 1
8 tg(x 1).
8. limn
7n + 9
7n 2
2 7n6
.
9.
f (x) =x2 2x 63
||x 9||(x2 + x 42).
10.
f (x) =
3x7, x ( ; 7),
5x2 + 8x + 8, x (7; 8],
8x8, x (8; +) .
11. f (x) =4x2 + x + 51
x2 x 12.
. 128 473 2012/2013 .
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12. f (x) =3 4x 3 6x
3 4x + 4 6x.
. 129 473 2012/2013 .
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MA , 1, 086
1. a) , ,
{xn} =
3 3n 5
10 3n + 8 a =
3
10,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
8n2 + 6
7n3 6n2 6n 2.
3. limn
4n 3
2
n2 5
2n
2 + 1n
2 + 612
.
4. limn
2 2n 4n3 + 5
4n3 2 2n 3.
5. limn
1 + n + 6n2
6n + 13 9n + 5n2
5n 9.
6. limx2
x3 8
x4 16.
7. limx0
3x2cos7x 6 tg3x
7 tg x + sin(5x2).
8. limn
n 6
n 5
9n8
.
9.
f (x) =x2 7x 18
||x 9||(x2 3x 10).
10.
f (x) =
3x+9, x ( ; 9),
3x2 + 5x + 9, x [ 9; 3),
3x2, x [3; +) .
11. f (x) =2x3 2x2 + 9x + 10
x2 + x 2.
. 130 473 2012/2013 .
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12. f (x) =2 2x 2 3x
2 2x 5 3x.
. 131 473 2012/2013 .
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MA , 1, 087
1. a) , ,
{xn} =
3 5n
3 5n + 3
a = 1,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
4n + 5
2.
3. limn
2n
2 8n + 7n
2 632
n6 26
2n 12
n2 n + 7.
4. limn
3n6 + 5 4n + n + 2
2n4 4 5n 4n.
5. limn
4 + 3n 8n2
2n + 27 + 7n 4n2
4n 4.
6. limx 2
x4 16
2x2 3x 14.
7. limx0
7 tg( 7x) 8x2cos8x
sin( 7x) + tg( 8x2).
8. limx1
xx+1x1 .
9. f (x) =x + 2
||x2 + x 2||.
10.
f (x) =
2x+1, x ( ; 1),
x2 + 8x + 8, x ( 1; 9),
6x3, x (9; +) .
11. f (x) =5x3
x2 1.
. 132 473 2012/2013 .
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12. f (x) =3 3x 5 5x
5 3x + 2 5x.
. 133 473 2012/2013 .
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MA , 1, 088
1. a) , ,
{xn} =
4 3n 7
11 3n + 7
a =4
11,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
6n 6
n2 7n 2.
3. limn
n 1
3n
2 6 3n + 8
3n
3 6n2 + 3n
2 + 512
.
4. limn
5 5n + 4 4n + 4
3 5n + 5 4n 2.
5. limn
1 7n 5n2
8n + 52 + 3n 2n2
n 3.
6. limx3
x2 + 3x 18
x4 81.
7. limx0
3 sin( 5x) + 6x2sin1x
5 sin9x + sin( 4x2).
8. limn
3n + 1
3n 4
4n+9
.
9.
f (x) =x2 10x + 16
||x 2||(x2 2x 48).
10.
f (x) =
1x+2, x ( ; 3],
x2 3x 1, x ( 3; 2),
1x+3, x [ 2; +) .
11. f (x) =x3 + 6x2 4x + 3
x2 2x + 1.
. 134 473 2012/2013 .
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12. f (x) =2 4x + 2 5x
4 4x 5 5x.
. 135 473 2012/2013 .
-
MA , 1, 089
1. a) , ,
{xn} =
10 3n + 4
9 3n + 8
a =10
9,
.
) , =1
100000000 N,
n > N ||xn a|| < .
2. limn
n2 + n 4 + 4n + 1
3n + 4.
3. limn
3n
3 + 2n2 + 7n + 3n6n 3 cos
9( 5n + 6)6n + 3
.
4. limn
5n3 2 6n
3n3 + 5 6n.
5. limn
9n
2 + 3n 4 3n .
6. limx2
x4 16
5x2 + x + 18.
7. limx
( x + 1)2
6x 8tg
7
6x + 8.
8. limn
8n 5
8n + 7
9 8n6
.
9.
f (x) =x2 14x + 48
||x 6||(x2 7x 8).
10.
f (x) =
3x7, x ( ; 4),
4x2 + 7x 1, x ( 4; 2),
5x+7, x [ 2; +) .
11. f (x) =x3 + 3x2 6x 8
x2 x 2.
. 136 473 2012/2013 .
-
12. f (x) =2 4x + 2 5x
5 4x + 2 5x.
. 137 473 2012/2013 .
-
MA , 1, 090
1. a) , ,
{xn} =
7 5n + 5
6 5n 7 a =
7
6,
.
) , =1
10000000 N,
n > N ||xn a|| < .
2. limn
5n16 + 3
4n14 + 9n + 7.
3. limn
n 2n 4 n 4
4
3n + 4
n
2 5n 22n 8
2 .
4. limn
4n3 3 7n + n
4n5 4 3n + 4n.
5. limn
5 3n + 2n2
3n + 44 + 8n n2
5n 8.
6. limx4
x5 1024
x2 2x 8.
7. limx
( x + 7)2
9x + 3tg
1
x + 3.
8. limx
6x6 2
6x6 + 8
2x62.
9. f (x) =||x 4||
x2 2x 8.
10.
f (x) =
7x4, x ( ; 3],
x2 + 4x + 4, x (3; 6),
8x5, x (6; +) .
11. f (x) =2x2 + 4x
x3 + 9x2 + 20x.
12. f (x) =3 3x + 2 5x
5 3x 3 5x.
. 138 473 2012/2013 .
-
MA , 1, 091
1. a) , ,
{xn} =
4n 4
n + 8 a = 4,
.
) , =19
1000 N,
n > N ||xn a|| < .
2. limn
8n18 + 8n6 + 1
3n14 3n9 + 1.
3. limn
n 2n + 73n + 4
2n + 3
2
n + 8
2 2n 4
3 .
4. limn
4n7 2 5n
3n6 + 2 7n + 2n + 2.
5. limn
7 4n + n2
n 48 7n + 4n2
4n 7.
6. limx 3
x
