春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,

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1 春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 32, n a n {a n } {a n } 2. a n = 10n + 1 {a n } lim an = n a n 9, 19, 29, 39, 49, n a n {a n } {a n } lim a n = n a n 0 1

2 春期講座 ~ 極限 2 3. a n = ( 2) n {a n } 2, 4, 8, 16, 32, 64, n a n a n lim α α {a n }, {b n } α, β lim a n = α, lim b n = β 1. lim ka n = kα k 2. lim (a n ± b n ) = α ± β 3. lim a nb n = αβ 4. β 0 lim a n b n = α β 2

3 春期講座 ~ 極限 3 {a n }, {b n } b n lim = 0 a n b n a n b n lim = α α a n b n a n a n = n 2 + n, b n = n lim a n =, lim lim b n = b n a n = n n 2 + n = 1 n + 1 b n a n = 0 (1) lim a 1 n = lim = 0 a n (2) lim a 1 n = +0 lim = a n (3) lim a 1 n = 0 lim = a n (4) {a n }, {b n } n (a n ) > (b n ) 1 b n lim = 0 a n b n a n lim a n = +0 {a n } lim a n = 0 {a n } (5) {a n }, {b n } n (a n ) = (b n ) a n b n (1 < r ) 1 (r = 1 ) (6) lim rn = 0 ( 1 < r < 1 ) (r 1 ) b n a n lim = α α 0 α a n b n 3

4 春期講座 ~ 極限 4 1 (1) a n = 1 n + 3 (4) a n = n2 + 1 n 3 1 (7) a n = n n3 n (2) a n = n 2 n (3) a n = 2n + 1 3n 1 (5) a n = n 1 n 2 (6) a n = n2 3n + 1 n 2 + 2n 2 (8) a n = n n 1 (9) a n = n n 2 + 2n (1)0 (2) (3) 2 3 (4)0 (5) (6)1 (7)0 (8) 3 4

5 春期講座 ~ 極限 5 2 (1) a n = 2n 3 n (2) a n = 4 1 n (3) a n = 2n 3 n 1 (4) a n = 3n + ( 2) n 1 3 n 2 n+1 (1) 5

6 春期講座 ~ 極限 6 3 (1) lim a n =, (2) lim a n =, (3) lim a n = α, lim b a n n = lim = 1 b n lim b n = lim (a n b n ) = 0 lim (a n b n ) = 0 lim b n = α (4) lim a n =, lim b n = 0 lim a nb n = 0 (1) a n = n, b n = n 2 a lim n = 0 b n (2) a n = n + 1, b n = n lim (a n b n ) = 0 (3) (4) a n = n 2 1, b n = n 2 + n 1 lim anbn = 1 6

7 春期講座 ~ 極限 7 r n a n a n = r n {a n } (1 < r ) 1 (r = 1 ) lim rn = 0 ( 1 < r < 1 ) (r 1 ) r n b n {a n }, {b n } lim = 0 a n a n b n b n lim = α α α a n lim 3n =, lim 3 n 5 n = lim lim 5n = ( ) 3 n = n 3 n a n b n {a n }, {b n } n (a n ) > (b n ) a n, b n b n lim = 0 a n 7

8 春期講座 ~ 極限 8 4 (1) 2n n + 3 (5) n n2 + 1 n (2) n 2 n (3) (6) log 3 (n + 2) log 3 n n + 1 n (4) 3 n 3 n + 2 n (1)2 (2) (3)0 (4)1 (5)1 (6)1 8

9 春期講座 ~ 極限 9 5 (1) lim 1 + r 2n (r ±1) (2) lim 1 r2n 1 r n + r n+1 1 r n (r 1) + rn+2 0 ( 1 < r < 1) (1) (r < 1, 1 < r) 1 (r = ±1) { 1 (r < 1, 1 < r) (2) r+1 1 ( 1 < r 1) 9

10 春期講座 ~ 極限 10 (1) A A x x a a A A x a x a A (2) {a n } a n {a n } a n a n = 1 n n a n 1 {a n} a n = n + 1 n a n = 1+ 1 n n a n 1 {a n } {a n } a 1 a 2 a 3 a n {a n }, {b n } lim a n = α, lim b n = β α, β n a n b n α β α n {a n }, {b n }, {c n } ˆ ˆ n b n a n c n lim b n = lim c n = α lim a n = α 10

11 春期講座 ~ 極限 11 {a n }, {b n } a n b n lim b n = lim an = ˆ ˆ n a n b n lim b n = lim a n = 11

12 春期講座 ~ 極限 12 6 (1) n a n < 0 lim n = 0 (2) n a n > 1 lim n = 1 (1)a n = 1 n (2)a n = n + 1 n 12

13 春期講座 ~ 極限 13 7 (1) a n = sin n n (2) a n = 1 + cos n n (3) a n = ( 1)n n (4) a n = 1 n sin nπ 2 (1)0 (2)0 (3)0 (4)0 13

14 春期講座 ~ 極限 14 8 (1) n 3 n > n 2 n (2) lim 3 n (1) (2)0 14

15 春期講座 ~ 極限 15 9 r > 1 (1) h > 0 (1 + h) n 1 + nh (2) (1) lim rn = (1) (2) 15

16 春期講座 ~ 極限 16 {a n } + a 1 + a 2 + a a n + a n n=1 a n = a 1 + a 2 + a a n + n=1 n S n = a 1 + a 2 + a a n lim S n = α α {S n } α α {S n} n (n + 1) + S n S n = n (n + 1) = n k=1 1 k (k + 1) k 1 k (k + 1) = 1 k 1 k + 1 S n = ( ) ( ) ( ) ( n 1 ) = 1 1 n + 1 n + 1 ( lim S n = lim 1 1 ) = 1 n a + b = b + a

17 春期講座 ~ 極限 17 ˆ ˆ 2

18 春期講座 ~ 極限 18 1 (1) (2) (3) (2n 1)(2n + 1) + 1 n n n=1 k=1 k (k + 1)! 3

19 春期講座 ~ 極限 19 2 (1) 1 + ( 1) ( 1) + (2) (3) (4) ( 1 1 ) ( ) ( ) + 4 4

20 春期講座 ~ 極限 20 {a n } : a 1, a 2, a 3,, a n, a 1 = a r S n na (r = 1 ) S n = a 1 rn (r 1 ) 1 r {r n } : r, r 2, r 3,, r n, 1 < r < 1 0 a, r {a n } 1 < r < 1 a 1 + a 2 + a a n + a 1 r ( = 1 ( ) ) r ˆ ˆ 1 < r < 1 S n a + ar + ar 2 + ar ar n 1 + a = 0 1 < r < 1 5

21 春期講座 ~ 極限 21 3 (1) (2) (3) (4) (5) n=1 n=1 5 3 n ( 3) n 2 2n+1 a n+1 a n n=1 6

22 春期講座 ~ 極限 22 4 f(x) = (1 x 2 ) + x(1 x 2 ) + x 2 (1 x 2 ) + + x n 1 (1 x 2 ) 7

23 春期講座 ~ 極限 23 1 (1) lim (2) lim (3) lim n 2 + 2n + 1 n 3 + n n + 1 n 2 n 4 n 3 n + 4 n (1)0 (2) (3) 1

24 春期講座 ~ 極限 24 1 (1) lim (2) lim (3) lim n 2 + n + 1 2n 2 + n + 1 ( n2 + 2 n) 3 n + 2 n 3 n+1 1 (1) 1 2 (2)0 (3) 1 3

25 春期講座 ~ 極限 25 2 ( ) lim 2 n cos nπ 3 3 0

26 春期講座 ~ 極限 26 2 lim 1 n sin nπ 3 0

27 春期講座 ~ 極限 27 1 (1) 1 < r < 1 (2) r = 1 (3) r < 1, 1 < r { r n } + 1 r n + 2 (1) 1 2 (2) 2 3 (3)1

28 春期講座 ~ 極限 28 3 (1) (2) (3) n=1 n=1 n=1 1 (n + 2)(n + 3) 1 2n + 1 2n 1 n + 1 2n + 1 (1) 1 3 (2) (3)

29 春期講座 ~ 極限 29 4 (1) (2) (1)3 (2)

30 春期講座 ~ 極限 30 3 (1) ( ) 2 ( ) 3 (2) (1) 2 3 (2)

31 春期講座 ~ 極限 31 2 A 1 B 1 C 1 A 1 B 1 C 1 A 2 B 2 C 2 A 2 B 2 C 2 A 3 B 3 C 3 A n B n C n S n S 1 + S 2 + S

32 春期講座 ~ 極限 32 5 (1) lim x 2 + 2x 3 x 1 x 2 1 x (2) lim x 1 x 1 (1)2 (2) 1 4

33 春期講座 ~ 極限 33 6 (1) lim x (2) lim x 3x 2 2x 1 2x 2 3x 2 1 x2 + 2x x (1) 3 2 (2)1

34 春期講座 ~ 極限 34 4 (1) lim x 2 x 2 x 2 x 2 3x + 2 x (2) lim x 0 x (1)3 (2) 1 4

35 春期講座 ~ 極限 35 5 (1) lim x 2 + 2x 1 x x + 1 ( (2) lim x2 + 3x x) x (1) (2) 3 2

36 春期講座 ~ 極限 36 3 a, b lim x 0 a x + 9 3b x = 1 a = b = 6

37 春期講座 ~ 極限 37 7 lim x 1 x sin x 0

38 春期講座 ~ 極限 38 6 lim x 0 x cos 1 x 0

39 春期講座 ~ 極限 39 8 (1) lim x +0 (2) lim x 0 x 2 x x x 2 x x (1) 1 (2)1

40 春期講座 ~ 極限 40 7 (1) lim x 2+0 (2) lim x 2 0 (3) lim x 2+0 (4) lim x x 2 1 x 2 x 2 2x x 2 x 2 2x x 2 (1) (2) (3)2 (4) 2

41 春期講座 ~ 極限 41 4 x = 0 [a] a (1) y = [x] (2) y = [cos x] (1) (2)

42 春期講座 ~ 極限 42 9 f(x) = 1 x (1) x = 1 (2) f (x) (1) 1 2 (2) 1 2x x

43 春期講座 ~ 極限 43 8 f(x) = x + 1 (1) x = 1 f(x) (2) f(x) f (x) (1) (2) 1 2 x + 1

44 春期講座 ~ 極限 (1) y = x x 1 x 2 (2) y = (3x 2 2x + 1)(4x + 3) (3) y = 2x2 1 x (1) 4x4 + 3x 2 x + 4 2x 3 (2)36x 2 + 2x 2 (3) x(2x3 3x 4) (x 3 + 1) 2

45 春期講座 ~ 極限 45 9 (1) y = (x 3 + 2x + 1)(x 2 1) (2) y = 2x 1 x (1)5x 4 + 3x 2 + 2x 2 (2) 2x2 + 2x + 2 (x 2 + 1) 2

46 春期講座 ~ 極限 46 5 f(x) = x (1) f(x) x = 0 (2) f(x) x = 0 (1) (2)

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a

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