《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.5 Infinitesimal and Infinite Limit

Chapter1FunctionsLimitsand$ 1.5 Infinitesimal and InfiniteLimit

Chapter 1 Functions and Limits §1.5 Infinitesimal and Infinite Limit

L.Infinitesimal1.DefinitionDef:We say that f (x) is an infinitesimal as x -→a(or x -→o) iflim f(x) = 0(or lim f(x) = 0)e-S Def:f(x) isan infinitesimalas x→a ifV>0,38>0, s.t.0<x-a<8=f(x)<S 1.5 Infinitesimal and Infinite Iimit

§1.5 Infinitesimal and Infinite limit I. Infinitesimal Def: We say that f (x) is an infinitesimal as if lim ( ) = 0 (or lim ( ) = 0) → → f x f x x a x x → a(or x → )   0,  0,s.t.0  x − a    f (x)    − Def: f (x) is an infinitesimal as if x → a 1. Definition

I. InfinitesimalQ:Infinitesimalisavery small number.O is an infinitesimal.An infinitesimalis a bounded function.is an infinitesimal.X2. The relationship between the infinitesimaland the limitTh: lim f(x)= A f(x)=A+α (α→0 as x→a)x>aS1.5 InfinitesimalandInfinite Iimit

§1.5 Infinitesimal and Infinite limit Q: ⚫ Infinitesimal is a very small number. ⚫ 0 is an infinitesimal. ⚫ An infinitesimal is a bounded function. ⚫ is an infinitesimal. x 1 I. Infinitesimal 2. The relationship between the infinitesimal and the limit Th: =  = + → f x A f x A x a lim ( ) ( ) ( → 0 as x → a)

II. Infinite Limit1. DefinitionDef: We say that lim f(x) = oo ifVM>0,38>0, s.t. v0 Myty= f(x)M2.Geometricinterpretation---0xaa+oSa-MS 1.5 Infinitesimal and Infinite Iimit

§1.5 Infinitesimal and Infinite limit y a − a a +  x o y = f (x) M − M II. Infinite Limit Def: We say that if =  → lim f (x) x a M  0,  0,s.t.0  x − a    f (x)  M 1. Definition 2.Geometric interpretation

Il. Infinite Limit1.DefinitionDef: We say that lim f(x) = oo ifVM>0,38>0, s.t. v0MJty= f(x)M2.Geometricinterpretation0xa+oaM01S 1.5 Infinitesimal and Infinite Iimit

§1.5 Infinitesimal and Infinite limit y a − a a +  x o y = f (x) M − M II. Infinite Limit 1. Definition 2.Geometric interpretation Def: We say that if =  → lim f (x) x a M  0,  0,s.t.0  x − a    f (x)  M

Il. Infinite LimitQ: olim f(x) = co =→ f(x)is a very large number.-lim f(x) = oo = f(x)is not bounded.X→It its converse true?y=xsinx3.Relation between the two definitionsTh: If f(x) →0 as x → a(x →8),1then>0asx→a(x→8)f(x)S 1.5 Infinitesimal and Infinite Iimit

§1.5 Infinitesimal and Infinite limit y = x sin x II. Infinite Limit 3. Relation between the two definitions Th: 0 as ( ). ( ) 1 then If ( ) as ( ), → → →  →  → →  x a x f x f x x a x Q: lim f (x) f (x) x a =   → is a very large number. lim f (x) f (x) x a =   → is not bounded. It its converse true?

II. Infinite Limit4.RelationtoAsymptotesHorizontalAsymptoteTheliney=aisahorizontal asymptoteofthegraphofy=f(x)if either lim f(x)=a or lim f(x)=a .Xx→+0VerticalAsymptoteThe line y = a is a vertical asymptote of the graph of y = f(x)if eitherlim f(x) = o0 or lim f(x)=oo :S1.5 InfinitesimalandInfinite Iimit

§1.5 Infinitesimal and Infinite limit 4. Relation to Asymptotes II. Infinite Limit Vertical Asymptote The line y = a is a vertical asymptote of the graph of if either =  → + lim f (x) x a or lim ( ) =  . → − f x x a y = f (x) Horizontal Asymptote The line y = a is a horizontal asymptote of the graph of if either lim f (x) a . x = →−  f x a or x = →+  lim ( ) y = f (x)

II. Infinite LimitExample1Findthe vertical and horizontal asymptotes of the graph2xof f(x)=x-12x2xSolutionlimlim+8-8x-1t x-1x-1 x-1So x =1 is a vertical asymptote.2x= 2limx-→0 x-1So y=2is a horizontal asymptote.S 1.5Infinitesimal and Infinite limit

§1.5 Infinitesimal and Infinite limit II. Infinite Limit Example 1 Find the vertical and horizontal asymptotes of the graph of . 1 2 ( ) − = x x f x = +  − → + 1 2 lim 1 x x x = −  − → − 1 2 lim 1 x x x So is a vertical asymptote. x = 1 2 1 2 lim = → x − x x Solution So is a horizontal asymptote. y = 2