《函数的极限2》PPT课件.ppt

14 函数的极限 二、自变量趋于无穷大时函数的极限 一、自变量趋于有限值时函数的极限 极限的通俗定义、极限的几何意义、 极限的局部保号性、 极限的精确定义、 左右极限 极限的通俗定义、极限的精确定义、极限的几何意义、 水平渐近线 一、自变量趋于有限值时函数的极限 自变量的变化趋势： x x0，x x0-0，x x0+0，x ，x -，x + f (x)=A或f (x) A(当x x0) 函数极限的通俗定义： 在自变量的某个变化过程中，如果对应的函数值 f(x)无限接 近于某一确定的常数A，那么这个确定的常数A就叫做在这一 变化过程中函数f(x)的极限当x x0时，f(x)以 A为极限记为 分析： 当xx0时，f(x) A当|x-x0| 0 时，|f (x)-A|能任意小 任给e 0， 当|x-x0|小到某一时刻，有|f (x)-A|0， 存在d 0， 使当|x-x0| 0， d0， x：00， d0， 使当00，d = 0， 所以 分析： |f(x)-A|=|(2x-1)-1|=2|x-1|，为了使|f(x)-A|0 ， d =e 0 ， 所以 只需 |x-1|0(或f(x)0， 取正数e A，根据极限的定义， 对于这个 取定的正数e ，必存在着一个正数d ，当00 点x0的某一去心邻域，当x在该邻域内时，就有f(x)0(或f(x)0(或f(x)0， d0， x： x0- d 0， X 0， x: |x|X，有 | f (x)-A|X的一切x， 对应的函数数值f(x)都满足不等式 |f(x)-A|X时，要证存在正数X， 分析：设e是任意给定的正数 因为对e0，X= ， 使当|x|X时，有 水平渐近线： 直线y=0是函数y = 的图形的水平渐近线 已知 x y O1 1 如果 ， O x y p 2 p 2 y=arctan x 例如，函数y=arctanx的图形的水平渐近线有两条： 则直线y=c是函数y=f (x)的 图形的水平渐近线 一般地， 和 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