InfinitesimalandInfinitetheembryonicformofthelimitconceptThatwhichissogreatthatnothingliesbeyonditiscalledtheGreatOne;thatwhichissosmallthatnothingcanbecontainedwithinitiscalledtheSmallOne.《UnderHeaven.AllUnderHeaven》"無外"meansthatitisimpossibleforanythingtobeoutsideofit.Ifthereisanoutside,itisnottheutmostgreatness,therefore,theutmostgreatnessisinfinite."無內(nèi)"meansthatitisimpossibleforanythingtobeinsideofit.Ifthereisaninside,itisnottheutmostsmallness,therefore,theutmostsmallnessisinfinitelysmall.Definingtheutmostgreatnessby'havingnooutside.'Definingtheutmostsmallnessby'havingnoinside.'infiniteinfinitesimalinfinitesimalinfiniterelationshipconclusionInfinitesimalandInfinite

Inotherwords:aninfinitesimalisafunction(orvariable)thatapproaches0astheindependentvariableundergoessomechange.1.InfinitesimalDefinition1InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion1.4無窮小與無窮大

4Answer:Thelimitoftheconstant0.00001isstill0.00001,whichisnotequalto0.Question(1):Is0.00001infinitesimal?Note1:Anynonzeroconstant(nomatterhowsmallitsabsolutevalue)isnotaninfinitesimal.0istheonlyinfinitesimalconstant.Zhuangzi·UnderHeaven:ThatwhichissosmallthatnothingiswithinitiscalledtheSmallOne.InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion1.4無窮小與無窮大

5

Note2:Infinitesimalmustbelinkedtothechangingprocessoftheindependentvariable.Wecannotsaythatavariableisinfinitesimalinisolation.solution:

∴

whenx→2,(x?1)isnotinfinitesimal.(x?1)isinfinitesimalwhenx→1.InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion

Forexample.InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion

theorem

1(Relationshipbetweeninfinitesimalsandfunctionlimits)Theinfinitesimalquantityinthesamechangeprocessoftheindependentvariableisdenotedasα.InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusionWhenx→x_0(orx→∞),ifthecorrespondingfunctionvalue|f(x)|increasesinfinitely,thenthefunctionf(x)issaidtobeinfinitewhenx→x_0(orx→∞).Thatis:infinityisafunction(orvariable)whoseabsolutevalueincreasesinfinitelyduringacertainchangeprocessoftheindependentvariable.2.InfiniteDefinition2

InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion2.Positiveandnegativeinfinite1)Whenx→x_0(x→∞),ifthefunctionvaluef(x)isalwayspositiveanditsabsolutevalueincreasesinfinitely,

2)Whenx→x_0(x→∞),ifthefunctionvaluef(x)isalwaysnegativeanditsabsolutevalueincreasesinfinitely,

example:

thenthenInfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion

Thefunction1/xbecomesinfiniteasx→0.example:Q:Is10^8infinite?zhuangzi:thatwhichissogreatthatnothingliesbeyonditiscalledtheGreatOneNote1.Anyconstantcannotbeinfinite.Infinityisastatethatdescribesafunction.Note2.Ifafunctionisinfinite,itmustbeunbounded.Buttheoppositeisnottrue!InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusionas

isinfinitesimal,and

then

isinfinite.as

isinfinite,

isinfinite;thenAccordingtothistheorem,allproblemsaboutinfinitycanbetransformedintoinfinitesimalsfordiscussion.Inthesameprocessofchangeoftheindependentvariable,note:3.TherelationshipbetweeninfinitesimalandinfinityDefinition2

InfinitesimalandInfiniteinfinitesimalinfiniterelationshipconclusion1.Theconceptofinfinitesimalandpointstonote2.Theconceptofinfinityandpointstonote3.Therelationshipbetweeninfinitesimalandinfinity:

infinitesimalinfiniterelationshipconclusionsummaryfunctionimageLimitExistenceBoundednessin(–∞,+∞)

。

in(0,1)

。Thinkingquestion:Please