Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2015
Math 1a, 2015 problem set 7, due Monday,November 23 10:00 A.M.
Do problems 1 and 2 from section 5.1 of Cranks , problems 1 from section 5.2 of
Cranks, problems 1 and 2 from section 5.3 of Cranks. (Be sure and refresh Cranks before
you do this. Chapter 5 m
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2015
1
Problem 2.1.5
Use the squeeze theorem to calculate
(
)
3
4+ 2 .
n
lim n
n
)
3
4+ 2 .
n
(
Solution. Let
an = n
We have
(
2+
3
4n
)2
4+
3
n
3
3
2+
2+ .
4n
n
Hence if we dene
(
)
3
3
bn = n 2 +
2 =
4n
4
then an bn for all n. Similarly, we have
(
)2
3
1
3
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2015
1
Problem 3.3.1
Since f is O(1), there are > 0 and a constant C so that if h < then f (h) < C. Since the denition of
limit as h 0 only depends on values of h smaller than , we have
lim
h0
f (h)g(h)
Cg(h)
lim
h0
h
h
g(h)
= C lim
h0 h
=0
since g is o(h).