第六节极限的存在准则与两个重要极限 习题 1.计算下列极限 (1) lim inax (2)Iim√xcot I-0 tan Bx 3)lim 3"sin I 1-cos 2x (4) lim (5)Im OSx x→∞2x+cosX 解(1)若a≠0,li lim snax os Bx= lim inax. Br.axa x-0 tan Bx I-0 sin Bx x-o ax sin BxBx B 若a=0,易知lim sinai (2) lim vxcot - A cos x= lin s√x=1 (4) lim √2 cOS x (6)lim x→∞2x+ costs→,COSx 2.计算下列极限: (1)lim(1+ax)x(a,b>0); (2)lim() x+11 第六节 极限的存在准则与两个重要极限 习 题 1-6 1. 计算下列极限: (1) 0 sin lim ( 0) x tan x x α β → β ≠ ; (2) 0 lim cot x x x → ＋ ; (3) π lim 3 sin 3 n n n→∞ ; (4) 0 1 cos 2 lim x sin x → x x − ; (5) 0 lim x 1 cos x x → + − ; (6) sin lim x 2 cos x x →∞ x x − + . 解 (1) 若α ≠ 0 , 00 0 sin sin sin lim lim cos lim xx x tan sin sin x x xxx x x x x xx α α α β αα β →→ → β β α ββ β = ⋅ = ⋅ ⋅= ; 若α = 0 , 易知 0 sin lim 0 x tan x x α α → β β = = . (2) 0 0 00 lim cot lim cos lim lim cos 1 x x xx sin sin x x xx x x → → →→ x x = ⋅= ⋅ = ＋ ＋ ＋＋ . (3) π sin π 3 lim 3 sin lim π π 3 π 3 n n n n n n →∞ →∞ = ⋅= . (4) 2 0 0 1 cos 2 2sin lim lim 2 x x sin sin x x → → xx xx − = = . (5) 0 00 2 lim lim lim 2 2 1 cos 2 sin sin 2 2 x xx x x x x x x → →→ + ++ = = ⋅= − . (6) sin 1 sin 1 lim lim 2 cos 2 cos 2 x x x x x x x x x x →∞ →∞ − − = =− + + . 2. 计算下列极限： (1) 0 lim(1 ) ( , 0) b x x ax a b → + > ; (2) 1 lim ( ) 1 x x x →∞ x − + ;