第七节无穷小与无穷大 习题1-7 1.利用等价无穷小替换定理求下列极限 (1) lim (2) lim sin(r") (m,n∈N); x→0(sinx)" (3)lim x(1-cos x) (4) lim arcsin 3x sin x x→0sin2x (5) lim arctan (6) lin 解(1)lin tan sx =lim x→02x2 (2) lim sin(r” 00=n<m (3)lim x(I-cos x) rx2 二 lir sIn x (4) lim arcsin 3x x→0sin2xx→+02x2 lim x -= lim = lim x x→0cosx-1 2.当x→0时,试确定下列无穷小关于x的阶数: (1) x+sinx (2)x32+10x2 (3)1-cos2 (4)tan 2x21 第七节 无穷小与无穷大 习 题 1-7 1. 利用等价无穷小替换定理求下列极限: (1) 0 tan 5 lim x 2 x → x ; (2) 0 sin( ) lim ( , ) (sin ) n m x x m n → x ∈ * N ; (3) 3 0 (1 cos ) lim x sin x x → x − ; (4) 0 arcsin 3 lim x sin 2 x → x ; (5) 1 lim arctan x x →∞ x ; (6) 1 2 3 0 (1 ) 1 lim x cos 1 x → x + − − . 解 (1) 0 0 tan 5 5 5 lim lim x x 2 22 x x → → x x = = . (2) 0 0 0 , sin( ) lim lim 1 , (sin ) . n n m m x x n m x x n m x x n m → → ⎧ > ⎪ == = ⎨ ⎪∞ < ⎩ 当 当 当 (3) 2 3 3 0 0 1 (1 cos ) 1 2 lim lim x x sin 2 x x x x → → x x ⋅ − = = . (4) 0 0 arcsin 3 3 3 lim lim x x sin 2 2 2 x x → → x x = = . (5) 1 1 arctan 1 lim arctan lim lim 1 x xx 1 1 x x x x x x →∞ →∞ →∞ = == . (6) 1 2 2 3 0 0 2 1 (1 ) 1 2 3 lim lim cos 1 3 1 2 x x x x x x → → + − = =− − − . 2. 当 x → 0 时, 试确定下列无穷小关于 x 的阶数： (1) x + sin x ; (2) 3 2 x +10x ; (3) 2 1 cos 2 − x ; (4) 2 tan 2x