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()