14 CHAPTER 1. LIMIT n+c+Vn+d n2+bn+c 4. Vn+avn+b 11.√n2+an+b-√n2+cn+d 12. n+avn+b-vn+cVn+d 6.vn(vn+a-√n+b) n2+n+ 7.√n+c(√m+a-n+b) n+a 8.Vn+a+n+b-2√n+c n2+bn +c 9. an+b n2+n+1 n2+cn +d Exercise 11.20. Find the limits 1.√n+ asm n-√h+ b cos n. 5.√m+(-1)"(√m+a-√h+b) n+ sinn Vn+bcos n 6.√m2+an+sinn +bn+cos n 3.,/m+(-1)ya an+sinn n+(-1)mb n2+bn+cosn 4. Vn+a+sinn n2+an+b mn+c+(-1)n n+(-1 Exercise 11.21. Find the limits 3.n2(3n+a-n+b a 4. y/n(vn+a-vvn+b) Exercise 11.22. find the limits 5.4 1./n 2. n+1 Exercise 1.1. 23. Find the limits /n+ bcos 2n 2.(m2+(-1)nc 3.(m2+bm+c Exercise 1. 1.24. Suppose limn-oo n 1. Use the arithmetic rule and the sandwich rule to prove that, if In< 1, then limn-oo zn=l. Of course we expect the condition In<1 to be unnecessary. See Example 1.1.2114 CHAPTER 1. LIMIT 3. √ n + a + √ n + b √ n + c + √ n + d . 4. √ n + a √ n + b √ n + c √ n + d . 5. √ n + a + b √ n + c + d . 6. √ n( √ n + a − √ n + b). 7. √ n + c( √ n + a − √ n + b). 8. √ n + a + √ n + b − 2 √ n + c. 9. r n n2 + n + 1 . 10. r n + a n2 + bn + c . 11. √ n2 + an + b − √ n2 + cn + d. 12. √ n + a √ n + b − √ n + c √ n + d. 13. n √ n2 + n + 1 . 14. n + a √ n2 + bn + c . 15. r n 2 + an + b n2 + cn + d . Exercise 1.1.20. Find the limits. 1. √ n + a sin n − √ n + b cos n. 2. r n + a sin n n + b cos n . 3. s n + (−1)na n + (−1)nb . 4. √ n + a + sin n √ n + c + (−1)n . 5. p n + (−1)n( √ n + a − √ n + b). 6. √ n2 + an + sin n − √ n2 + bn + cos n. 7. r n 2 + an + sin n n2 + bn + cos n . 8. √ n2 + an + b n + (−1)nc . Exercise 1.1.21. Find the limits. 1. √3 n + a − √3 n + b. 2. 3 r n + a n + b . 3. √3 n2( √3 n + a − √3 n + b). 4. √3 n( p3 √ n + a − p3 √ n + b). Exercise 1.1.22. Find the limits. 1.  n − 2 n + 15 . 2.  n − 2 n + 15.4 . 3.  n − 2 n + 1− √ 2 . 4.  n + a n + b p . Exercise 1.1.23. Find the limits. 1.  √ n + a sin n √ n + b cos 2n p . 2.  n 2 + an + b n2 + (−1)nc p . 3.  n + a n2 + bn + c p . Exercise 1.1.24. Suppose limn→∞ xn = 1. Use the arithmetic rule and the sandwich rule to prove that, if xn ≤ 1, then limn→∞ x p n = 1. Of course we expect the condition xn ≤ 1 to be unnecessary. See Example 1.1.21