epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

102 The Nuts and Bolts of Proof, Third Edition
x^ - 1 3
3. Prove that lim -
x-^ix^ — 1 2*
4. The choice of 8 is not unique. In Example 1, we proved that when
£ = 4.5, we can use 8 = 1.5. Show that if one chooses 8 = 0.9, it is still
true that |(3x - 5) - l|<4.5.
5. Prove that in Example 1 one would have |(3x — 5) — l| <4.5 for all
values of x with \x — 2\<8 for any 8 < 1.5.
6. Prove that lim = 0.
n^oo 3n + 1
7. Prove that lim -^—- = 0.
n->oo n -\- 1
o T. 1 ,. 5n+l 5
8. Prove that lim = -.
n->oo 3n — 2 3
9. In Example 4, we determined that when s = i, N4/5 = 2.75. Prove
|2n-l 14
that if we use M4/5 = 16, then ;— 2 < - for all n>M4/s.
/ I n +1 I 5 /
an = —T- \ whose limit is zero. Given
^^ \n=l .
n+ 1
8>0, prove that the statement —;; 0 <s is true for all n such
I ^ I
that: (a) n>Ns = ;^(l + VTT4^\, (b) n>Ms = ^-^,
(This exercise is meant to reinforce the fact that for a given s one can
have several choices for the number N.)