例5求lm tan 5x-cosx +1 sin x R. tanx=5x +o(r), sin 3x= 3x+o(r), 1一c0sx=。x2+0(x2) 5x+0(x)+x2+0(x2) 原式=lim 2 x→0 3x+o(x) 0(x oX 5+ -X十 x 2 m ox 53 3+例 5 . sin 3 tan 5 cos 1 lim0 x x x x - + → 求 解 tan x = 5 x + o(x), sin 3x = 3x + o(x), ( ). 21 1 cos 2 2 - x = x + o x3 ( ) ( ) 21 5 ( ) lim 2 2 0 x o x x o x x o x x + + + + = → 原式 x o x x o x x x o x x ( ) 3 ( ) 2 ( ) 1 5 lim 2 0 + + + + = → . 35 =