数学分析(一)期中试卷答案.pdfVIP

数学分析（一）期中试卷答案 一．1.A; 2.A; 3.C; 4.B; 5.B; 6A; 7.B; 8.D; 9.A; 10.D. 二．(用数学符号叙述正确的每个给一分,叙述错误的每个给0 分) （1） lim f (x ) ≠A x →−∞ 解：There exist a positive number ε ,for any positive number G,we can 0 find (there exists) x satifying the inequality x −G , such that the inequality f (x ) −A ≥ε0 holds. (2) lim f (x ) =−∞ x →x +0 0 解：Let the function f (x ) be defined in the right semi-neighborhood of x0 , If for each positive number G, no matter how large ,it is possible to find a positive number δ , such that for all values of x different from x0 and satisfying the codition 0 x −x0 δ , we have the inequality f (x ) −G . 1 1 ⎡ ⎤ 三．（1）lim −⎢ ⎥ 1 x → x x ⎣ ⎦ n ⎧1 1 1 x , 1 1 ⎪⎪ −n n +1 n ⎡ ⎤ x Proof: 1. ∵ −⎢ ⎥ ⎨1 1 1 (2 分) x x ⎣ ⎦ ⎪ − − x ⎪⎩x (n 1), n n −1 1 1 1 ⎡ ⎤ n (n 1) 1 x 0 − → − − ∴ → + , ⎢ ⎥ n x x ⎣ ⎦ 1 1 1 ⎡ ⎤ − → − x → −0 , ⎢ ⎥ n n 0 (5 分) n x x