《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.8 Comparison of Infinitesimals

Chapter1Functions and Limits$ 1.8 Comparison of Infinitesimals

Chapter 1 Functions and Limits §1.8 Comparison of Infinitesimals

I.Example(x-1)?x-1x?-1sinxlimlimlimlimx-1 (x-1)2x-1x→1x-1x→0x-1x=2=0=1=8$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals 1 1 lim 2 1 − − → x x 1 x ( 1) lim 2 1 − − → x x x x x x sin lim →0 2 1 ( 1) 1 lim − − → x x x I. Example = 0 =  = 2 = 1

Il.DefinitionAssume that α→0,β→0.β(1) if lim=0,thenβisahigherorderinfinitesimalofαα(denotedby β = o(α)B(2) if limA0,thenβandαareofthesameorderαβ(3) if lim-l,βandαare equivalentinfinitesimalsα(denotedby α ~ β);βif lim(4)=A, thenβis akthorder infinitesimal of α.Qf$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals II. Definition Assume that  → 0, → 0, ； ， ( ( )) (1) lim 0 .       = o = denotedby if then is a higher orderinfinitesimal of (2) if lim 0, then  and are of the same order;   = A  (4) lim   .   if A then is a kth orderinfinitesimal of k = ， ( ~ ); lim 1, ,       denotedby (3) if = and are equivalent infinitesimals

Il.DefinitionForinstancelim= 0,then, x2 = o(3x) (x →0).3xx→0sinxlim1.then,sinx ~ x (x→0)x→0xwhen x →0,sinx~ x,tanx-sinxtanx~x, 1-cosx~1tanx-sinx is athird order infinitesimal of x$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals II. Definition when x → 0, 0， 3 lim 2 0 = → x x x 1， sin lim 0 = → x x x (3 )( 0). then, x 2 = o x x → then,sin x ~ x (x → 0). For instance 2 3 2 1 , tan sin ~ 2 1 sin x ~ x, tan x ~ x, 1− cos x ~ x x − x x tan x −sin x is a third order infinitesimal of x

III. TheoremTh1 α~ β β=α+o(α)asx→0, sinx ~x,1-cosx2sinx = x+o(x)1-cosx=-x2VX2y= sin x=1.cosx1231$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals III. Theorem sin x = x + o(x) ( ) 2 1 1 cos 2 2 − x = x + o x as x → 0, y = 1 − cos x 2 2 1 y = x 2 2 1 sin x ~ x, 1− cos x ~ x y = x y = sin x Th1  ~    =  + o()

IlI. TheoremTh2If α(x) ~ α,(x), β(x) ~ β,(x) as x →xo, and there existsa(x)a(x)f(x)α(x)f(x)limThen, limlimx-→xo β,(x)β(x)β,(x)x→xox→xo$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals III. Theorem . ( ) ( ) ( ) lim ( ) ( ) ( ) lim 1 1 0 0 x x f x x x f x x x x x     → → Then, = If (x) ~ 1 (x), (x) ~ 1 (x) as x → x0 , andthere exists Th2 , ( ) ( ) lim 1 1 0 x x x x   →

Example:Findthefollowinglimitstan 2x1-cosxtanx-sinx(1) lim(2) lim(3) limsin'xx→0 sin5xx-→0x-0xsinx2x2?x-x= lim2= lim= limx→0 5xtx-→0x-→0x.x213X52= lim2tsx-→012$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals Example： Find the following limits. x x x x x x x sin 1 cos (2)lim sin5 tan 2 (1) lim 0 0 − → → x x x 5 2 lim →0 = . 5 2 = x x x x  = → 2 0 2 1 lim. 2 1 = 3 0 lim x x x x − = → x x x x 3 0 sin tan sin (3)lim − → 3 3 0 2 1 lim x x x→ = . 2 1 =

Example:Find the following limits/1+x -1arcs(4) lim x(1 - cos =(5) lim(6) limx-→0x-0x-→0xxx$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals Example： Find the following limits. x x x x x x x x x arcsin (6)lim 1 1 ) (5)lim 1 (4)lim (1 cos 0 0 2 →0 → → + − −

ExercisesJimn->0x + arctan x元2. lim3. lim sin(-x)tan 3x6元x-→o x-arctanxX6/1+xsinx-cosxlimx-→0xsinx$1.8Comparison of Infinitesimals

§1.8 Comparison of Infinitesimals Exercises ) 1 ) (1 3 1 )(1 2 1 lim(1 2 2 2 n - - - n  → 1. x x x x x arctan arctan lim − + → 2. x x x )tan 3 6 lim sin( 6 − →   3. x x x x x x sin 1 sin cos lim 0 + − → 4