嘉应数学分析课件9577812

1、 2 f ( x), g( x) 6.5 () a , bf(x) , g(x) a, b (i)(ii)f(x) , g(x) (a, b) ;(iii)f c2 ( x)gc2 ( x) ! 0 ;g(a) z g(b) .(iv) () , (a , b) f c( )gc( )f (b)f (a).g(b)g(a) 俆 f , x g ug( x),vf (x). O- uv . ) , ( dv AB ():dux f (b)f (a).kABg(b)g(a)vP( g( ), f ( )xxB( g(b),f (b)xA(g(a) ,f (a)Ou f (b)f (a)(g(x)F

2、(x)f ( x)f (a)g(a).g(b)g(a), F ( x) , (a , b),F c( ) f c( )0, f (b)f (a) gc( )0.g(b)g(a)gc( ) z 0( f c( ) f c( ) (iii) f (b)f (a) .g(b)g(a), gc( ) a, b(a 0) , (a, f, (a , b) , bf c( )ln.f (b)f (a)a a, b(a 0) , (a, f, (a , b) , bf c( )ln.f (b)f (a)ag(x) a, b ln x , f (x), g( x), (a , b), f c( ) , f (

3、b)f (a)1lnb.lna , , ) . . . ., 1.0 06.6 f g lim g( x)xo x0(i)lim f ( x)xo x00 ;D x0 U( x0 )(ii) gc( x) z 0 ;f c( x)gc( x)rf,limxo x0f(iii) A.Af c( x)gc( x)f (x)limxo x0limxo x0A.g( x) 0, f , gf ( x0 )g( x0 ) x0 . x U D ( x0 ), x0 , x( x, x0 ) f c( )gc( )f ( x)f ( x)f ( x0 )(x x ).0g( x)g( x)g( x )0

4、x o x0 , o x0 , f c( )gc( )f c( x)gc( x)f ( x)limxo x0limxo x0limxo x0A.g( x) 1 x o x0 x o xx o0 x 0x of,x of, . lim 1tan x . sin4 xxo 4 lim 1tan x . sin4 xxo 4傼 0 .0sec2 xlim 1tan x21 .limsin 4 x4cos 4 x42xo xo 44f c( x)gc( x)00 , limxo x0 f c( x) . limgc( x)xo x01lim ex- (12x) 2 . 2ln(1x)xo01lim e

5、x- (12x) 2 . 2ln(1x)xo0 xx2 ) x2 , 12x)2ln(11- (1- (1exex2x)2limxo0limx2 )2x)x2(1ln(1xo0- 12- 32- (1exex2x)limlim1.2 x2xo0xo0 .x. limxo0x1e .x. limx1exo0 0 , 0, . tx, x o 0, t o 0xt1limxo0limt o0limt o01.etet1x1e- 3xx3200. lim ()- x32xx1xo1- 3xx3200. lim ()- x32xx1xo13x230 lim()0- 2 x6 xxo1 3 x213 .

6、2limxo1 6x2 Sarctan x( 0 )02. lim 1xxo f Sarctan x( 0 )02. lim 1xo fx1x2x211 limlim1.2xo f 1xxo fx2sin1xx20() 0 limxo0xe1sin1xx20() 0 limxe1xo02 x sin 1cos 1lim xxexxo0sin1x2lim x xsin1 lim x1u 00ex- 1xo0 ex - 1xxo0x o 0sin1x1,cos,x of,sin x,cos xxL.Hospital lim tan xx . x2tanxxo0 lim tan xx . x2tan

7、xtan xxxo0limsec2 x1 lim23 x3xxo0xo0lim 2sec2 x tanx1 limtan x1 .6 x3x3xo0xo0 lim tan xx . x2tanxtan xxxo0limsec2 x1 lim23 x3xxo0xo0lim 2sec2 x tanx1 limtan x1 .6 x3x3xo0xo0 2. f f6.7 f g lim g( x)f(i)lim f ( x)xo x0xo x0 0 x U D ( x(ii)0 ) gc( x ) z 0;lim f c( x )A,rf, f(iii)A.gc( x )xo x0f c( x)gc

8、( x)f ( x) limxo x0limxo x0A.g( x) A . x0 H ! 0 x1 U D ( x0 ),xx1 x,f c( x)gc( x)H,A x, x1, f c( ) .f ( x1)f ( x)gc( )g( x1 )g( x) f c( )gc( )f (x1 )f ( x)H,AA(1)g( x1 )g( x) g( x1 )1f ( x)f (x1 )f ( x)f ( x1)f ( x)g( x)f ( x1 )1.g( x)g( x1 )g( x)g( x1 )g( x)1f (x)(1), GG! 0, x0xx0x1 ,; H ! 0, x 1 x

9、 o,x0G1 , x0G1 ! 0, x0xH ,f (x)f ( x1)f ( x)(2)g( x)2g( x1 )g( x)G1 (1) (2), x0xx0 x f ( x)H,Ag( x) f ( x)limxo x0A.g( x)ff f, ,A x o x 0 x o x x0 o x0x of,x ofx o f . . lim ln x . xxo f x o x 0 x o x x0 o x0x of,x ofx o f . . lim ln x . xxo ff. f1xln xlimxo flim0.x1xo fex lim3 .xo f xex lim3 .xo f

10、xexexexlim exf.limlimlim32xo f 6x6xo f xxo f 3xxo fex lim3 .xo f xexexexlim exf.limlimlim32xo f 6x6xo f xxo f 3xxo flim2xsinx. xof 2 xsin xex lim3 .xo f xexexexlim exf.limlimlim32xo f 6x6xo f xxo f 3xxo flim2xsinx. xof 2 xsin xf ,. flim 2xsin xlim 2cos x .(3)xof 2 xsin xxof 2cos x lim 2cosx,xof 2cos

11、 xsin x2lim 2xsin xlim x1.xof 2sin xxof 2 xsin xx(3) . :f cxfxlim limxof gxgcxxof: A arctan x .lim xo farctan2x, lim arctan2x, 2 lim arctanxxo f2xo f A = 1. 4x2arctan x1 1limlim2,x2xo f arctan 2 x2xo f 1. : 0 f .享俆 f0 3. 0 f1f00f0f r f, f .0 f0 .1.0 f 僔: 0 f1 f f 0 f 0 1 0.,ff00 0 ff0. f2.ff 110 00

12、.僔:f000(0 f ) lim x ln x.xo0 (0 f ) lim x ln x. xo0ln x xln x, 1x1lim ln xx- 1lim x ln xxo0limxo0lim(xo0x)0. 1xo0xx2(0 f ) lim x ln x. xo0ln x xln x, 1x1lim ln xx- 1lim x ln xxo0limxo0lim(xo0x)0. 1xo0xx2 :x1lim xln2xo0lim xln xxo0limxo0limxo0x11ln x, , x ln2x.- 1 ).x1( ff) lim(sin xxo0- 1 ).x1( ff)

13、lim(sin xxo0 lim xsin xlim 1cos x0.x sin xxo0xo0 sin xxcos x3.00 ,1f ,f0000ln0 fln1 1fo0 lnf 0f.f0 1( 1f ) lim(cos x) x2 . xo 01( 1f ) lim(cos x) x2 . xo 0 1 lncos x x2lncos x002.x, limxo 0(cos x)ex2 lim lncos x sin x lim1 ,x2xo0 2 x cos x2xo01 21-lim(cos x)xe2 .xo 01 lim x1x .( 1f )xo11 lim x1x .( 1f)xo11 ln xx lime1xo1lim ln xexo11 x1lim xexo1 11.e lim xx .xo0( 00 ) lim xxxo0.( 00 )ln xlimx o0 limx ln xex o01xlim ex ln x xo0e1 x 1 x2limxo0e0e1.1( f0) lim (cot x)ln x .xo0 1( f0) lim (cot x)ln x .xo0 1 (cot x)ln x 1 ln(cot x )eln x, 1 1 1ln xcot xsin2 x1 ln(cot x) limxo0limxo0x elim x?1.1,co