微积分_例题解-8

例8.讨论函数y=f(x)x= xx≥0 xx0+△x△x→>0+△x △x △x f(o=lin Ax Ax>0-△x△x>0-△x△x0-△x f(0)≠f(0), 所以函数y=x在x=0处不可导 上页 结束 下页

上页 结束 下页 解 例 8 讨论函数    −   = = = 0 0 ( ) | | x x x x y f x x 在 x=0 处的连续性 与可导性 因为 lim | | lim 0 0 0 = = → + → + x x x x  lim | | lim ( ) 0 0 0 = − = → − → − x x x x  所以 lim | | (0) 0 0 = = → x f x  从而函数y=|x|在x=0处连续 因为 lim 1 | | (0) lim lim 0 0 0 =   =   =    = + + → +  →  →  + x x x x x y f x x x  lim 1 | | (0) lim lim 0 0 0 =−  − =   =    = − − → −  →  →  − x x x x x y f x x x  (0) (0) + − f   f   所以函数y=|x|在x=0处不可导 lim | | lim 0 0 0 = = → + → + x x x x  lim | | lim ( ) 0 0 0 = − = → − → − x x x x  lim 1 | | (0) lim lim 0 0 0 =   =   =    = + + → +  →  →  + x x x x x y f x x x lim 1 | | (0) lim lim 0 0 0 =   =   =    = + + → +  →  →  + x x x x x y f x x x  lim 1 | | (0) lim lim 0 0 0 =−  − =   =    = − − → −  →  →  − x x x x x y f x x x lim 1 | | (0) lim lim 0 0 0 =−  − =   =    = − − → −  →  →  − x x x x x y f x x x 