1. 1. LIMIT OF SEQUENCE nta 4.m+(-1)a n+(-1)2b 1)b n+bsin n 1 (-1)"(an+b) √n2+an+b n2+c(-1)n+1n+d 3. Vn+c+d an+ bsin n n2+an+b 1)ny2⊥d cn+dsin n Exercise 1.1. 15. Find the limits, p>0 sin 2.sin(n+1) asin n+b 3. mp nP+(-1 on np+c Example 1. 1.5. For a>0, the sequence vn+a-vn satisfies a 0<√n+a (n+a-vn)(n+a+n) n+a+vn n+a+√nn By mn* v分0 and the sandwich rule, we get limn→(Vm+a-√n)=0. Similar Argument also shows the limit for a <0 Example 1.1.6. The sequence Vn- satisfies 1<,n+2n+2 1+2 n By limn=oo(1+2n)=1+2. 0=1 and the sandwich rule, we get limn+ooV n= Exercise 1. 16 Show that limn-+oo(Vn+a-vn)=0 for a<0 Exercise 1.1.17. Use the idea of Example 1.1.5 to estimate 1 and then find n+2 limn→ Exercise 1.1.18. Show that limn-ooV n+6 1. You may need separate argument for a>b and a b Exercise 1.1.19. Find the limits 1.√m+a-vn+b /nta n+c+√n+1.1. LIMIT OF SEQUENCE 13 1. √ n + a n + (−1)nb . 2. 1 √3 n2 + an + b . 3. √ n + c + d √3 n2 + an + b . 4. n + (−1)na n + (−1)nb . 5. (−1)n (an + b) n2 + c(−1)n+1n + d . 6. (−1)n (an + b) 2 + c (−1)nn2 + d . 7. cos √ n + a n + b sin n . 8. cos √ n + a √ n + b sin n . 9. an + b sin n cn + d sin n . Exercise 1.1.15. Find the limits, p > 0. 1. sin √ n np . 2. sin(n + 1) np + (−1)n . 3. a sin n + b np + c . 4. a cos(sin n) np − b sin n . Example 1.1.5. For a > 0, the sequence √ n + a − √ n satisfies 0 < √ n + a − √ n = ( √ n + a − √ n)(√ n + a + √ n) √ n + a + √ n = a √ n + a + √ n < a √ n . By limn→∞ a √ n = 0 and the sandwich rule, we get limn→∞( √ n + a − √ n) = 0. Similar argument also shows the limit for a < 0. Example 1.1.6. The sequence r n + 2 n satisfies 1 < r n + 2 n < n + 2 n = 1 + 2 1 n . By limn→∞  1 + 2 1 n  = 1 + 2 · 0 = 1 and the sandwich rule, we get limn→∞ r n + 2 n = 1. Exercise 1.1.16. Show that limn→∞( √ n + a − √ n) = 0 for a < 0. Exercise 1.1.17. Use the idea of Example 1.1.5 to estimate r n + 2 n − 1 and then find limn→∞ r n + 2 n . Exercise 1.1.18. Show that limn→∞ r n + a n + b = 1. You may need separate argument for a > b and a < b. Exercise 1.1.19. Find the limits. 1. √ n + a − √ n + b. 2. √ n + a √ n + c + √ n + d