2.2函数的极限

2.2函数的极限,函数极限问题是研究当自变量,趋向于,的变化趋势,或趋向于无穷大时，函数,（两种基本变化趋势）,自变量的变化,趋向于无穷,趋向于一点,1、自变量趋于正无穷时函数极限,例1,证,2、自变量趋于负无穷时函数极限,例1,证,3、自变量趋于无穷时函数极限,证,几何解释:,2、自变量趋于有限值时函数的极限,xx22.00004.0000000001.50002.2500000001.30001.6900000001.00001.0000000000.80000.6400000000.50000.2500000000.50000.2500000000.30000.0900000000.20000.0400000000.10000.0100000000.08000.0064000000.06000.0036000000.04000.0016000000.02000.0004000000.01000.0001000000.00100.0000010000.00010.000000010,例1,例2,证,就有,故,例3,证,注意：,左极限,右极限,(2).单侧极限,左极限和右极限统称为单侧极限.,函数当时极限存在的充分必要条件是左极限及右极限各自存在并且相等,即,即使和存在,但不相等,则不存在.,例4函数,解,而右极限,因为左极限和右极限存在但不相等，所以,不存在.,关于极限定义中的去心邻域,xx22.00004.0000000001.50002.2500000001.30001.6900000001.00001.0000000000.80000.6400000000.50000.2500000000.50000.2500000000.30000.0900000000.20000.0400000000.10000.0100000000.08000.0064000000.06000.0036000000.04000.0016000000.02000.0004000000.01000.0001000000.00100.0000010000.00010.000000010,-2.00004.000000000-1.50002.250000000-1.30001.690000000-1.00001.000000000-0.80000.640000000-0.50000.250000000-0.50000.250000000-0.30000.090000000-0.20000.040000000-0.10000.010000000-0.08000.006400000-0.06000.003600000-0.04000.001600000-0.02000.000400000-0.01000.000100000-0.00100.000001000-0.00010.000000010,二、函数极限的性质,性质1:（唯一性）,函数极限如果存在，则一定是唯一的.,性质2:（局部有界性）,性质3:（保号性）,性质