例5求lm tan 5x-cosx+1 x→0 sinr F tanx=5x +o(x), sin 3x=3x+o(x), 1-cosx=x2+0(x2) 2 5x+0(x)+x2+0(x2) 原式=lim 2 →0 3x+0(x) 0(x 0(x 5+ lim x2x 5 x→>0 0(x 3 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 =