文档介绍：函数极限(Limits of Functions).xxxsin时的变化趋势当观察函数???极限的概念极限定义(Definition of a limit)The number L is the limit of the function f(x) as x approaches c (or approaches infinity) if, as the values of x gets arbitrarily (but not equal) to c(or approaches infinity), values of f(x) approach (or equal) L. We write lim ( ) (or lim ( ) )x c xf x L f x L? ??? ?0andx x x? ??极限,即自变量的值无限趋近但不等于某规定数值(或自变量绝对值无限增大)时,函数值趋近于某个定值。设函数y=f(x)在点附近有定义(在点可以无定义),若当x无限接近于时,函数f(x)的值无限接近于常数A,则称当x趋于时,f(x)以A为极限,记作或f(x)→A(x→)0x0lim ( )x xf x A??0x0x0x0x(描述性定义)1. x→x0时函数的极限Consider the function as 2: ( ) f x x? ??x→2(from the left)……(x)xf(x)x2lim ( ) 5xf x???2lim ( ) 5xf x???2 the left-hand limit equals the right-hand limit,we can write: lim ( ) 5xIff x??This is called the left-hand limit and is written:This is called the right-hand limit and is written:………… ……x→2 (from the right)f(x)→5f(x)→500 000 0If lim ( ) lim ( ) , then llim (im) lim ( ) lim (())x xx xx xxx xx x xf x f x A f xf x f x A f x AA?????? ????? ? ?? ???极限存在定理极限存在定理设函数y=f(x)在点左(或右)侧有定义(在点可以无定义),若当x无限接近于时,函数f(x)的值无限接近于常数A,则称当x趋于时,f(x)以A为左(或右)极限,记作0 0x x? ?(或)0 0lim ( ) lim ( )x x x xf x A f x A? ?? ?? ?(或)0x0x0 0x x? ?(或)左极限与有极限统称为单侧极限(one-sided limit).左右极限又可分别记为0 00 0( 0) lim ( ) ( 0) lim ( )x x x xf x f x f x f x? ?? ?? ? ??与2. x→∞时函数的极限1Consider the function as : ( ) .x f xx?? ?If lim ( ) lim ( ) ,then lim ( ) .x xxf x f x Af x A??? ?????? ??1lim 0xx???★1lim 0xx????1lim 0xx????21