2.3 Inequalities and Limits 81( c) Calculate hm. (9 - + -^9 + --^9 + · · · + -9). What does this tell
n-><X> 10 100 1000 lOn
you about the decimal 0.9999999 · · ·?- Let c be a fixed real number. Prove that Vp EN,
(a) lim en = 0 if lei:::; 1, and (b) lim nP = 0 if lei> 1.
n-><X> nP n-><X> en
Prove that if 0 :::; a :::; b, then lim yl an + bn = b.
n->
Suppose {an} and {bn} are sequences such that a;+ b;-> 0. Prove that
an -> 0 and bn -> 0. Does this conclusion remain true if a; +b; is replaced
by a~ + b~? Justify your answer.
Finish the proof of Theorem 2.3.10, by explaining why lim lxnl = 0
n->
follows from lim lxn 1 +kl = 0.
k->
Prove Corollary 2.3.11. (Consider c = 0 as a separate case.) For c =f. 0,
use Theorem 2.3.10 and the first squeeze principle.
Prove Theorem 2 .3. 12 (b).
Prove that if x > 1, then lim !!:._ = 0.
n-+oo xn
k
Prove that if k E N 3 k ?: 2, then n-+oo lim nk n = 0.
Find each of the following limits, using the appropriate limit theorems to
justify your answer.
1+2n
(a) lim --
n_,= 3n
1+2n
(c) lim --
n->= 1 - 2n
( e) lim ( V n^2 + 2 - n)
n->
(b) lim 2n + 3n
n-><X> 5n(d) lim l - Vn
n->= 1-n
( f) lim ( V n^2 + n - n)
n-><X>(g) lim ( Jn+I -fa) (h) lim (.!. - -
1
-)
n->= n-><X> n n + 1
n2
(i) lim - 1
n-+oo n.n2
(j) lim -
n-><X> 2n- Prove Theorem 2.3.14.
- Show by example that Theorems 2.3.12- 2.3. 14 do not remain true if all
the inequalities are changed from :::; to <, or ?: to >. Which inequalities
can be changed without jeopardizing the truth of the theorems?