Advanced Calculus Assignment 1

1. Advanced Calculus, Assignment 1
1. Let f : R → R and c ∈ R. Show that
lim
x→c
f(x) = L if and only if lim
x→0
f(x + c) = L.
2. Let {yn}n∈N be a sequence in a metric space Y . Define a function on the
set
D =
1
n
: n ∈ N
by f 1
n = yn. Show that f has a limit at 0 if and only if {yn}n∈N is
convergent.
3. Using the definition of limits (the ”epsilon-delta” definition), verify the
following limits:
(a)
lim
x→3
(x2
− x) = 6
(b)
lim
x→c
x = c for all c ∈ R
(c)
lim
x→4
√
x = 2
(d)
lim
x→1
x
1 + x
=
1
2
4. Use the sequential criterion of limits to establish the following limits:
(a)
lim
x→2
1
1 − x
= −1
(b)
lim
x→9
1
√
x
=
1
3
(c)
lim
x→0
x2
|x|
= 0
(d)
lim
x→1
x
1 + x
=
1
2
1

2. 5. For p ∈ R, let f(p) = (2p + 1, p2
) ∈ R2
. Then clearly f : R → R2
. Use the
definition of limit to prove that
lim
p→1
f(p) = (3, 1)
with respect to the Euclidean metrics on each space.
6. Let c ∈ R and let f : R → R be such that limx→c(f(x))2
= L.
(a) Show that if L = 0, then limx→c f(x) = 0.
(b) Show by example that if L = 0, then f may not have a limit at c.
7. Determine the following limits and verify your result:
(a)
lim
x→1
(x + 1)(2x + 3)
(b)
lim
x→1
(x + 1)
(2x + 3)
(c)
lim
x→1
(x + 1) + (2x + 3)
(d)
lim
x→1
(x + 1) − (2x + 3)
8. Consider the functions f, g and h defined by
f(x) = x + 1 g(x) = x − 1 h(x) =
2 if x = 1
0 if x = 1
(a) Find limx→1 g(f(x)) and compare with the value of g(limx→1 f(x)).
(b) Find limx→1 h(f(x)) and compare with the value of h(limx→1 f(x)).
9. Let f : R → R and g : R → R be such that limx→c f(x) = L and
limx→L g(x) = M. Prove that limx→c g(f(x)) = M.
10. (a) Give an example of a function that has a left-hand limit but not a
right-hand limit at a point.
(b) Give an example of a function that has both left- and right-hand
limits, but not a limit at a point.
11. Define f : R → R by
f(x) =
x − 1 if x < 0
x + 1 if x ≥ 0
2

3. (a) If {xn}n∈N is an increasing (decreasing) sequence which converges
show that {f(xn)}n∈N is also an increasing (decreasing) sequence
which converges.
(b) Find an example of a sequence {yn}n∈N which converges but {f(yn)}n∈N
does not converge.
12. Find an example of a function f : R → R, a sequence {xn}n∈N and a point
c so that f(xn) → f(c) but xn → c.
13. Prove that the following limit does not exist
lim
x→0
f(x)
where f(x) = 1 if x is a rational number and f(x) = 0 if x is irrational.
14. Prove that the following limit does not exist:
lim
x→0
1
x
15. Let f(x) =
1 if x ∈ Q
0 if x ∈ R Q.
(a) Does limx→0 f(x) exist? Explain why.
(b) Does limx→0 f|Q(x) exist? Explain why.
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