MATH 1002 Fall & Winter 2016 Assignment 4 Due Thursday October 13 in class. Please include your tutorial section number or TA name on your assignment. Please STAPLE your assignments. The following questions are to be turned in to be graded. 1. (4 marks) Let p be a positive integer. You may assume without proof that np n for all n N. Use the precise definition of a limit diverging to + to prove that limn np = +. 2. Suppose k and ` are positive integers, and ak , ak1 , ..., a0 , b` , b`1 , ..., b0 R with ak 6= 0 and b` 6= 0. Define (sn ) by sn = ak nk + ak1 nk1 + ... + a1 n + a0 b` n` + b`1 n`1 + ... + b1 n + b0 (we assume that the denominator is not equal to zero for any n N). Use the various limit laws from section 9 to prove the following results. Please indicate which theorems you are using. ak . n bk (b) (5 marks) Prove that if k < `, then lim sn = 0. (a) (5 marks) Prove that if k = `, then lim sn = n ak > 0, b` (c) (7 marks) Prove that if k > ` and then lim sn = +. Be careful: n Theorem 9.10 can not be used here as some terms may be negative. 3. (4 marks) Exercise 9.10 (c). Use the precise definition of a limit in your proof, and not previously seen theorems. The following are suggested exercises and are not be to turned in. (i) What is lim np when p is a negative integer? Justify your answer. n (ii) Exercises 9.1, 9.2, 9.3, 9.5, 9.8, 9.9, 9.10 (a) and (b), 9.11, and 9.13 (iii) What is lim sn in question 2 when k > ` and n ak b` < 0? Justify your answer. \f