文档介绍：第一章函数与极限Chapter1 FunctionandLimit1映射与函数(MappingandFunction)一、集合(Set)二、映射(Mapping)映射概念(TheConceptofMapping)设X,Y是两个非空集合,如果存在一个法则/,使得对X屮每个元素x,按法则/,在丫屮有唯一确定的元素y与之对应,则称/为从X到Y的映射,记作LetX,Ybetwononomptysets,ifthereexistsarulefthatassociatesauniqueelementyofYtoeveryelement兀inX,thenfiscalledamappingfromXtoY,denotedbyfX-^、函数(Function)函数概念(TheConceptofFunction)设数集DuR,则称映射f:X^Y为定义在Q上的函数,通常简记为y=/(x),xe£>,其中x称为自变量,y称为因变量,D称为定义域,记作Df=DoLetthenumbersetDuR,thenthemapping/:XTYiscalledafunctiondefincdonD,usuallydenotedbyy=f(x),xEDfwherexiscalledanindependentvariable,yiscalledadependentvariable,Discalledadomain,denotedbyDf=(LimitoftheSequenceofNumber)一、数列极限的定义(DefinitionoftheLimitofSequenceofNumber)设{£}为一一数列,如果存在常数d,对于任意给定的止数£(不论它多么小),总存在正整数N,使得当吋,不等式|心-°|<£都成立,那么就称常数G是数列{£}的极限,或者称数列{£}收敛于d,记为limx?J=a或xna(n—>oo)on->o)Let{xn}beasequenceofnumber,Ifthereexistsaconstanta,suchthatforanygivenpositivenumber£,thereexistsapositiveintegerN,suchthatforeveryn>N,xn-a<€,thentheconstantaiscalledthelimitofthesequence,orwecallthatthesequence{xn}convergestoa,denotedbylimxn=aorxnTa{nToo)n—>oc二、收敛数列的性质(PropertiesofConvergentSequence)定理1(极限的唯一性)如果数列{£}收敛,那么它的极限唯一。Theorem1(UniquenessofLimit)Ifthesequence{xn}isconvergent,(收敛数列的有界性)如果数列{%}收敛,那么数列{£}必定有界。Theorem2(BoundednessofaConvergentSequence)Ifthesequence{xn]isconvergent,then{xnJisbounded・定理3如果0(或gvO),那么存在止整数N>0,当n>N时,都"TOO有X”>0(