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<x-a<8 =f(x)> 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. v0<x-a<8 =f(x)>MJty= 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