tailieunhanh - Calculs 9: Series, selutions

To help you have more learning materials and prep math, invite you to consult the document content "Calculs 9: Series, selutions". Content document provides you questions exercises and limited function. Hope this is useful references for you. | Calculus - 9. Series Solutions 1. Compute the limits. 1 a l 2a l 3a 1 lim---------------------------------- X- 0 X m lim X 0 .2 lim - ị m n e X-Slxm-1 . 2x - 3 20 3x - lim -----. X X 2a I 50 l 30 1 lim ------- 1 0 1 X f lim ------------ 1 0 1 X lim X-H 1 X i- i 1 x 2 Solution a Fora 0 we have 1 6 lla 6a 2. By Example 2 a V __ 1 1 2a l 3a 1 2 Zz 2 I lim ---------- --------------- lim 6 Ila 6a 2 6 Ila 6a 2 L o 6. b Let X 7 0. By the binomial formula we have with some polynomials p x and ợ aộ 1 mx n 1 nx m _ a 2 1 ma m2a 2 x3p x 1 na n2a 2 a 3ợ a a 2 n n 1 m2x2 m n2x2 x3 p x q x a 2 mnln m . XX ----- ----------F x p x - q x . Taking the limit X 0 in the above expression we obtain mn n m 2. c Since xn 1 x l a -1 a -2 H------ X 1 we have for X 1 xn - 1 _ a -1 a -2 X 1 xm 1 a m-1 xm 2 X 1 Since the rational function on the right hand side is defined at X 1 we obtain by Example 2 a Q-n _ 1 I n-2 I . I I 1 1 t -I- 1 t I I I I -I- Ờ x -1 xm 1 x -1 xm-1 xm 2 X 1 m d Since both polynomials p a 2a - 3 3a - 1 and i a 2a 1 are of degree 50 by Examples the limit lima 0Op x ợ x is ÍI50 Ồ50 where ữ50 and Ồ50 are the leading coefficients of p and q respectively. Hence p x 220330 3 30 x -oo ợ x 250 2 e We use the E-S characterization of the left-hand limit lim f x 00 given X a 0 in Homework b to show that 1 -------- oo- - l-0 1 X Given E 0 choose s 1 E then 1 0 1 X 0 E implies This proves the claim. f We use the E-S characterization of the right-hand limit lim f x 00 given X a 0 in Homework d to show that lim ------ 00. x-41 0 1 X Given E 0 choose s E then 1 Ẽ 0 X 1 Ổ implies E and finally X 1 1 X -E. This proves the claim. _______ g The limit does not exist since the left-hand and the right-hand limits do not coincide. h Given E 0 choose 5 1 ỰẼ. Then . 1 I 5 ựẼ implies that is 1 lim ------X 00. 1 - X 2 2. Using inequalities formulate the following statements and give an example for each. a lim f x b X OQ c lim f x 00 X 00 b lim f z oo X a 0 d lim fix 00 xDa Q Solution