.
{xn}, e N, xn , n>N,
êxn - a ê < e. (1.1)
: xn a.
(1.1)
a- e < xn < a + e, (1.2)
, x n, n>N, (a-e, a+e), .. e- .
, , , - .
, xn = f(n) n.
f(x) a - D(f), .. , D(f), a. a D(f), .
1. f(x) xa, {xn} , , {f(xn)} .
, .
2. f(x) xa, , e, d >0 ( e), x, d- , .. x,
0 < ½x-a½ < d, f(x) e- , .. êf(x)-A ê < e.
, e - d.
1 2 . f(x) x a , ,
f(x) = A. (1.3)
, {f(xn)} ( ) x , , f(x) , :
f(x) = ¥ ( f(x) = - ¥).
(.. ), , .
, , .
.
1. f(x)=A, g(x)=B,
(f(x)+(g(x)) = A + B, (1.4)
f(x) g(x) = AB, (1.5)
f(x)/g(x) = A/B (B ¹ 0). (1.6)
. 0/0, ¥ /¥, 0 × ¥, ¥ - ¥ , , , .
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2. (f(x))a = ( f(x)) a, a = const, (1.7)
.. , , ;
bf(x) =bA, b = const, f(x)=A; (1.8)
logc f(x) = logc f(x), c = const. (1.9)
3. = 1, = 1, a = const, a >0,
= 1, (1.10)
(1 + a)1/ a = e, (1.11)
e 2.7 - . (1.10) (1.11) .
(1.11):
= logc e, (1.12)
(aa - 1)/a = ln a, (1.13)
((1 + a) m - 1)/a = m, (1.14)
,
= 1.
E x a x > a, x a+0. , , a=0, 0+0 +0. xa x<a, xa-0. f(x) . f(x) xa , = .
f(x) x0,
. (1.15)
(1.15) :
,
, .
(1.15) , , x = xo f(x) . y = 1/x. R, x = 0. x = 0 D(f), , .. , 0, D(f), . f(xo)= f(0) , xo = 0 .
f(x) xo,
,
xo,
.
xo .
, xo, , , , -, , -, f(xo). , , .
1. f(xo), , f(x) xo , .
2. ¥ , , xo .
, y = ctg x x +0 , +¥, , x=0 . y = E(x) ( x) , .
, [a,b], [a,b]. .
, - . , , : , , , ..
. . , e . e e = . . , , . , . 100 . . 100 % . , 100 . . 200 .. , 100 . ., . 100 . . 100 × 1,5 = 150, - 150 × 1,5 = 225 (. .). 1/3 , 100 . . 100 × (1 +1/3)3237 (. .). 0,1 , 0,01 , 0,001 .. 100 . . :
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100 × (1 +1/10)10 259 (. .),
100 × (1+1/100)100 270 (. .),
100 × (1+1/1000)1000 271 (. .).
, , 271. 2,71 , 100% , , ,
= e.
1. , , xn =(n-1)/n , 1.
. , , e>0 , N, , n > N ½ xn -1 ½<e.
e >0. ½ xn -1 ½=½(n+1)/n - 1½= 1/n, N 1/n<e. n>1/e , , N 1/e, N = E(1/e). , xn = 1.
2. , xn = .
. . n ¥ , . xn, n2, n. , , :
xn = .
3. xn = . xn.
. = .
: .
4. ().
. , ¥ - ¥. :
= .
5. f(x)=21/x. , .
. 1 . { xn }, 0, .. xn =0. , f(xn)= -. xn = 1/n. , 1/n =0, = 2n = +¥. xn xn = -1/n, . = 2- n= 1/2n = 0. 2 1/x .
6. , sin x .
. x1, x2,..., xn,... - ,
xn = ¥. {f(xn)} = {sin xn } xn ¥?
xn= pn, sin xn= sin pn = 0 n sin xn =0.
xn=2pn+p/2, sin xn= sin(2pn+p/2) = sin p/2 = 1 n sin xn =1. , sin x .
7. .
. : = 5 . t = 5x. x0 : t0. (3.10), 5 .
8. .
. y=p-x. xp, y0.:
sin 3x = sin 3(p-y) = sin (3p-3y) = sin 3y.
sin 4x = sin 4(p-y) = sin (4p-4y)= - sin 4y.
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=- .
9. .
. arcsin x=t. x=sin t x0 t0. = .
10. 1) ; 2) ; 3) .
.
1. 1 , : .
, , 1 , : = .
2. , .. 0/0. . . x-2, x ¹ 2 :
= .
(x+1) ¹ 0, , ,
= = .
3. x¥ . . x2 :
= .
11. .
. : , x-90, .. .
, ,
.
12. .
. = .
6.2.
, .. , (, , . .). - . - , , . :
S = P(1 + i)n. (1.16)
P - , i - ( ), S - , n - . , . , , . , : S, n, P. , S , S - P . P, S, , , S. :
P = Þ P = = 0.
, .
- , . . , . - , , . ( ) , , , . , m :
S =P (1 + i/m) mn.
, m - , i - . m, . m ¥ :
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`S = P (1 + i/m) mn = P ((1 + i/m) m) n.
(1 + i/m) m = e i, `S = P e in.
- , . , , . , , d, `S = Pe .
d m¥. .
.
, , - , . , , . (), () . , , , . -, . . - , , . .
13. 1 . , , - 5% . 10 6 ´ 1,053 3 , 10 6 ´ 1,052, 2 . . , : 10 6 ´ 1,053; 10 6 ´ 1,052; 10 6 ´ 1,05; 10 6. . , . : S - , R - ,
i - ( ), n - ( ). n - 1, n - 2,..., 2, 1 0 ,
R (1 + i)n - 1, R (1 + i)n - 2,..., R (1 + i), R.
. (1+i) R. . : S = R´((1 + i)n - 1)/((1 + i) - 1) =
= R´((1 + i)n - 1)/ i. S n; i = ((1 + i)n - 1)/ i . m , S = R´((1 + i/m)mn - 1)/((1 + i/m) m - 1), i - .
a n; i = (1 - (1 + i) - n)/ i . n ¥ , :
a n; i = (1 - (1 + i) - n)/ i =1/i.
14. , - . - , .
,
, :
R´((1 + i)n - 1)/ i ¥ n ¥.
a n; i 1/i, A = R/i, . . .