1549901369-Elements_of_Real_Analysis__Denlinger_

2.2 Algebra of Limits 71

• 
  
  
  
  13. Suppose lim Xn = S and lim Yn = -3. Find each of the following:
      n-+oo n-+oo
      x2 - 2y2
      <(a) lim (2xn - 3yn)' (b) lim n n
      n-+oo n-+oo 4XnYn
  
  
  
  

_(c) lim lxn + 2ynl - (d) lim 1 + ,jx;;,

n-+oo Xn - Yn n-+oo y'l - Yn

14. Prove Theorem 2.2. 13 (f).

,;-lS. In each of the following, find the limit and use the "Algebra of Limits"
and other theorems of this section to justify your answer:

___ (-a) lim -

2

6

7

= (b) lim -

4

1

=

n-+oo + n n-+oo - Sn

{-c) lim _n_ =

n-+oo 3n + 8

~e) lim ( lOn - 11 ) 2

' n-+oo 7 - 2n

. 100
(g) hm - 2
r"'" n-+oo n

'--(") l l" Im f*s --=

n-+oo n^2 + s

(k) lim 3n2 + n - S

, _ n-+oo n^2 + 6n

,(m) lim

2

~n

n-+oo n - 7

Jo)

{q)

n^3 - 2n^2

lim

n-+oo 6 - 3n + 2n3

lim^1 + Vn =

n-+oo 3 - Jn

. 2n
(d) hm --=
n-+oo 3 - 4n

( f) lim ( S - 2n) 3

n-+oo 1+6n

. 6n
(h) hm 4 =
n-+oo n

(j) lim 8n + 9 =

n-+oo 8 - 3n2

r 4n^2 - Sn

~fl) n~~ 8n 2 + 3n - 1

3n^2 + 7n- 4

(n) nl~~ n3 - 4

2n^3 + 8n^2 - 23

(p) lim

n-+oo Sn3 - 6n

(r) lim ~ =

n-+oo 1 - n

16. Use mathematical induction to extend Theorem 2.2.13 (d) to prove that if

{xn} converges, then Vp EN, {x~} converges, and lim x~ = ( lim xn)P.

n-+oo n-+oo