10 CHAPTER 1. LIMIT The arithmetic rule is used in the second and third equalities. The limits limn-ooc 1 c and lim =0 are used in the fourth equalit Exercise 1.1.3. Find the limits 留++ (n+1)(n+2 2n3+3n 6 n2+3)3 (n+1)(n+2) Exercise 1.1.4. Find the limits ++ (√n+1)(√m+2) 3=切 2n-1 4n+1 (n+1)(n+ 9.( Exercise 1.1.5. Find the limits a +6 n+bvn+c (c√m+d) n (c√+d)3 n+a cn+d (a√n+b)2 m2+bn+c (√m+a)(√m+b) (cVn+d) Exercise 1.1.6. Show that 0. if lim …+a1n+a0 +bin+bo ifp= g and b≠0. bq Exercise 1.1.7. Find the limits 20 lo (2n+1)2-1 10m2-5 10n-5 Exercise 1.1. 8. Find the limits10 CHAPTER 1. LIMIT The arithmetic rule is used in the second and third equalities. The limits limn→∞ c = c and limn→∞ 1 n = 0 are used in the fourth equality. Exercise 1.1.3. Find the limits. 1. n + 2 n − 3 . 2. n + 2 n2 − 3 . 3. 2n 2 − 3n + 2 3n2 − 4n + 1 . 4. n 3 + 4n 2 − 2 2n3 − n + 3 . 5. (n + 1)(n + 2) 2n2 − 1 . 6. 2n 2 − 1 (n + 1)(n + 2). 7. (n 2 + 1)(n + 2) (n + 1)(n2 + 2). 8. (2 − n) 3 2n3 + 3n − 1 . 9. (n 2 + 3)3 (n3 − 2)2 . Exercise 1.1.4. Find the limits. 1. √ n + 2 √ n − 3 . 2. √ n + 2 n − 3 . 3. 2 √ n − 3n + 2 3 √ n − 4n + 1 . 4. √3 n + 4√ n − 2 2 √3 n − n + 3 . 5. ( √ n + 1)(√ n + 2) 2n − 1 . 6. 2n − 1 ( √ n + 1)(√ n + 2). 7. ( √ n + 1)(n + 2) (n + 1)(√ n + 2). 8. (2 − √3 n) 3 2 √3 n + 3n − 1 . 9. ( √3 n + 3)3 ( √ n − 2)2 . Exercise 1.1.5. Find the limits. 1. n + a n + b . 2. √ n + a n + b . 3. n + a n2 + bn + c . 4. √ n + a n + b √ n + c . 5. ( √ n + a)(√ n + b) cn + d . 6. cn + d ( √ n + a)(√ n + b) . 7. an3 + b (c √ n + d) 6 . 8. (a √3 n + b) 2 (c √ n + d) 3 . 9. (a √ n + b) 2 (c √3 n + d) 3 . Exercise 1.1.6. Show that limn→∞ apn p + ap−1n p−1 + · · · + a1n + a0 bqnq + bq−1nq−1 + · · · + b1n + b0 =    0, if p < q, ap bq , if p = q and bq 6= 0. Exercise 1.1.7. Find the limits. 1. 1010n n2 − 10 . 2. 5 5 (2n + 1)2 − 1010 10n2 − 5 . 3. 5 5 (2√ n + 1)2 − 1010 10n − 5 . Exercise 1.1.8. Find the limits