.,. f(),, f(),,.,. () lim 2 +3 2 9 (2) lim 3 3 2 9 (4) lim ( ) 2 3 +3 (5) lim 2 9 (6) lim + (7) lim (8) lim (9) lim (0) lim 2 3 + 3 9 2 2 +3 () lim sin 2 sin 2 (2) lim +3 () lim 2 2 9 = 5 5 = 3 (2) lim 3 2 9 = lim 3 = (4) lim ( ) = 2 +3 = 6
(5) lim 3 +3 (6) lim 0 (7) lim (8) lim (9) lim (0) lim 2 9 = lim 2 +3 9 = 3 = lim + = =0 2 3 + 3 9 = 2 2 +3= () lim sin 2 = sin 2 (2) lim =0 ++ = lim ( + +) = + ++ lim 2+/ 2 9/ =2.. sin lim =, lim ( + log ± ) = e, lim =.. () lim sin sin a (2) lim (b 0) sin b cos sin (4) lim( + ) / (5) lim( + ) / (6) lim( + a) / (a 0) log( + + 2 ) (7) lim 2 e (8) lim () lim sin = (2) lim sin a sin b = lim sin a a b a sin b b = a b 2
cos sin = lim 2(sin /2) 2 (sin /2)2 (/2) 2 = lim 2 sin (/2) 2 sin = 2 (4) lim( + ) / = e. = 0 t. t ( (5) lim( + ) / = lim ) ( + ) / = ( e (6) lim( + a) / = lim ) ( + a) /a a = e a log( + + 2 ) log( + + 2 ) + 2 (7) lim = lim = 2 + 2 2 2 e (8) lim =. = log( + t) t 0. =0. () f() = 2 +3 { 2 +3 ( 0) (2) f() = ( =0) { sin ( 0) (3) f() = 0 ( =) sin ( ) ( 0) (4) f() = 0 ( =0) { ( 0) (5) f() = 0 ( =0) () lim f() =3,f(0) = 3 =0 (2) lim f() =3,f(0) = =0 f() =,f(0) = =0 (4) lim f() = n =/2nπ n, n = /(2n +)π n. f(0) = 0 =0 (5) lim f() =. f(0) = 0 =0 =0. 3
2 sin ( ) ( 0) () f() = 0 ( =0) ( + ) sin ( ) ( 0) (2) f() = 0 ( =0) ( 0) (3) f() = +e / 0 ( =0) () f() 2 lim f() =0. f(0) = 0 =0 (2) lim f() =. f(0) = 0 =0,>0 f() = 0, lim =0,<0 f() =. lim f() =. f(0) = 0 f : R R, (i) f( + y) =f()+f(y) (, y R) (ii) f() =, f() =., {a n } lim n a n =. (i) n f(n) =f( + + + ) =nf() = (ii) f(n) =nf() = n =n. = m n (m, n ) f(m) =f(n) =nf() =nf( m n ) f( m n )= n f(m) = n m = m n f() =. = y =0 (i) f(0) = 0, y = f(0) = f()+f( ) f( ) = f(). f() =. 4
f() R f() =. f : R R, (i) f( + y) =f()f(y) (, y R) (ii) f() = e, f() =e. ( ) 2 f() =f 0 f(). g() = log f() 2 g : R R. (i) g( + y) =g()+g(y) (, y R) (ii) g() =. g() =, f() =e. () ( 2)( 3) + ( 3)( 4) + ( 4)( 2) = 0 [2, 3], [3, 4], [4, 5]. (2) n n 00. (3) sin cos =0 (π, 3π/2). () f() =( 2)( 3) + ( 3)( 4) + ( 5)( 6) f(2) = 2 > 0, f(3) = < 0, f(4) = 2 > 0. (2) f() = n f(0) =, f(000) > 0.. (3) f() = sin cos f(π) =π>0, f(3π/2) = < 0.. 5
() ( 2 ) cos + 2 sin =0 [0, ]. (2) 2 sin =0. (3) a n > a n + a n 2 + + a + a 0, n, a n cos n + a n cos(n ) + + a cos + a 0 =0 (0, 2π) 2n. () f() =( 2 ) cos + 2 sin f(0) = < 0, f( π 3 )=(π2 9 ) 2 + 6 2 > 0. (2) f() = 2 sin 0 f(0) = 0 < 0, f(00) > 00 2 0 > 0.. (3) f() =a n cos n + a n cos(n ) + + a cos + a 0 f(0) = a n + a n + + a 0 > a n ( a n + a n 2 + + a + a 0 ) > 0 ( ) π f < a n +( a n + a n 2 + + a + a 0 ) < 0 n ( ) 2π f > a n ( a n + a n 2 + + a + a 0 ) > 0 n ( ) 2nπ f > a n ( a n + a n 2 + + a + a 0 ) > 0 n. C., ϕ C :[0, ] [0, ]. 6
[0, ] \C ϕ C. C, ( ) 3, 2 ϕ C () = 3 2, ( ) 9, 2 9 ϕ C () = 4, ( ) 7 9, 8 9 ϕ C () = 3 4, ( ) 27, 2 27 ϕ C () = 8, ( ) 7 27, 8 ϕ C () = 3 27 8, ( ) 9 27, 20 ϕ C () = 5 27 8, ( ) 25 27, 26 ϕ C () = 7 27 8. ϕ C (). ϕ C () [0, ] \C. [0, ] \C. C 0, ϕ C () [0, ]. y = ϕ C (), (0 ). y = ϕ C () 0. C =0,α α 2 α n C( [0, ]) (α n =0 α n =2). y = ϕ C (), C( [0, ]) y = ϕ C () y = ϕ C () =0.β β 2 β n [0, ] 7
. ( α n =0 β n =0, α n = β n = ) y = ϕ C () C. [0, ] \C,. y = ϕ C () (0 ) 2. n, y = ϕ n () ( [0, ]). ϕ n () ( ) 2 n + 3. ϕ C (). ( ) 2 2n + 3 ( ) 2 n ( 2 2n lim n + + 3 3) =2.,. f(, y),, f(, y),,.,. y +3 2 y 2 ( ) 2 (y ) 2 (y +2) 3 +3y y( 2 9) y + y y,y 0 2 +3y () lim 2,y (2) lim,y,y 0 (4) lim,y (5) lim,y (6) lim (7) lim,y 0 sin 2y 8
sin a (9) lim (b 0) sin b (0) lim ( + )y/,y 0 () lim,y 0 log( + y + 2 ) 2y () lim 2,y (2) lim,y,y 0 y +3 2 y = 5 2 3 ( ) 2 (y ) = 2 (y +2) 3 +3y y( 2 9) (y +2) 2 +3y = lim,y 0 y( 9) y>0, = y /4 (y +2)+3y /2 = lim y>0,y 0 y /2 (y /4 9) = (4) lim,y lim,y. y (5) lim,y (6) lim (7) lim,y 0 (8) lim,y 0 y + y = lim,y lim y>0,y 0 lim y>0,y 0 (y +2)y /2 +3y y(y /4 9) 2. 9y/2 y( ++y) + y 2 = 2 +( y 2 )/ =. = k( y2 ) =. y = k.,y 0 2 +3y = sin 2y =. y = k/. sin ay sin by = a b ( + (,y 0 )y/ = lim ) ( + ) / y =,y 0 (9) lim log( + y + 2 ) log( + y + 2 ) y + 2 = lim =. 2y,y 0 y + 2 2y y = k. (0) lim,y 0. y( +3) () lim 3,y 2 9y 2 9
(2) lim,y 0 y y,y 0 y y2 3 + (4) lim,y 0 3 9y 2 sin 2y (5) lim,y 0 y y( 3) () lim =. =3 0. y = 3,y 2 9y2 6 (2) lim,y 0,y 0 (4) lim,y 0 (5) lim,y 0. y = y y =. y 2 3 + 3 9y = 2 sin 2y y lim,y 0 y 2 + 2 =0 9y =. y = k/ 2 (0, 0). () f(, y) = y y 2 ((, y) (0, 0)) (2) f(, y) = 2 +y 2 0 ((, y) =(0, 0)) (3) f(, y) = y 2 +y 2 ((, y) (0, 0)) 0 ((, y) =(0, 0))., y 2 + y 2 0
() y 2 + y 2. (2) y = k (, y) (0, 0) lim f(, y) =0. f(0, 0) = 0 (0, 0) (,y) (0,0) lim f(, y) = k2 (,y) (0,0),y=k +k 2 (0, 0). (3) y 2 + y 2 + y 2 2 lim f(, y) =0. f(0, 0) = 0 (0, 0). (,y) (0,0) () f(, y) = + y (2) f(, y) = (3) f(, y) = y 2 2 +y 2 ((, y) (0, 0)) 0 ((, y) =(0, 0)) ( 2 y 2 ) 2 +y 2 ((, y) (0, 0)) 0 ((, y) =(0, 0)) () + y + y 2 2 + y 2 lim (,y) (0,0) f(, y) =0. f(0, 0) = 0 (0, 0). (2) y 2 + y 2, f(, y) y. lim (,y) (0,0) f(, y) =0. f(0, 0) = 0 (0, 0). (3) y = k (, y) (0, 0) k2 lim f(, y) = (,y) (0,0),y=k +k 2 (0, 0)., X, Y f : X Y. X O X = {O λ ; λ Λ} X O λ. (X, O X ) () (3)
() O X,X O X (2) Λ 0 Λ, Λ 0 (3) Λ 0 Λ λ Λ 0 O λ O X λ Λ 0 O λ O X. O X X, O O X. (X, O X ), (Y, O Y ). f : X Y a X, f(a) V Y V, a U X U f(u) V. X, f X. O Y O Y f (O Y ) O X. (X, O X ). X K K λ Λ, λ,,λ n Λ K O λ (O λ O X ) n O λi i=. X,. R,. (X, O X ). X O,O 2 O X. X = O O2,O O2 = 2
R (a, b) [a, b], A R, a, b A (a <b) [a, b] A.. (X, O X ), (Y, O Y ). f : X Y f(x) Y. f(x), O,O 2 O Y X = O O2 O O2 =. f (O ), f (O 2 ) X = f (O ) f (O 2 ) f (O ) f (O 2 )=. f f (O ), f (O 2 ) O X. X. f(x). ( ) y = f() [a, b] f(a) <f(b) f(a) <λ<f(b) c [a, b] f(c) =λ f([a, b]) [f(a), f(b)] f([a, b]) f(a) <λ<f(b) λ f(c) =λ c [a, b] (X, O X ), (Y, O Y ). f : X Y f(x) Y. f(x) O λ (O λ O Y ). f f (O λ ) λ Λ O X (λ Λ) X f ( λ Λ O λ )= 3 λ Λ f (O λ )
. X, λ,,λ n Λ n X f (O λi ) i= n f(x) O λi i=. f(x). y = f() [a, b]. [a, b],, 2 [a, b] f( ) = ma [a, b] f(), f( 2) = min [a, b] f(). f :[a, b] R, [a, b] f([a, b]) R.. f([a, b]) = [c, d] (c<d). d c., 2 [a, b] f( ) = ma [a, b] f(), f( 2) = min [a, b] f(). {f n ()}. {f n ()} I. I, {f n ()}. f(), f(). I, ɛ>0, n 0. n>n 0 f n () f() <ɛ 4
n 0, ɛ. n 0 ɛ. ɛ>0, n 0 n>n 0, I f n () f() <ɛ.. { n } [0, ],. [0, /2]. ( ), f() = { ( =) 0 (0 <)., (0, ) n>n 0 n 0 <ɛ. n> log ɛ log. n 0. [0, /2] n> log ɛ log /2 log ɛ log log ɛ, log /2 n 0. R, ( + 2 ) n. [, ]. ( ), f() = { ( =0) 0 ( 0) 5
., 0 n>n 0 0 <ɛ ( + 2 ) n log ɛ. n> log( + 2 ). n 0. [, ] n> log ɛ log( + ) log ɛ log( + 2 ), log ɛ log 2 n 0.,. 6