MONASH UNIVERSITY
DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS
ETC2440-BEX2440
GROUP ASSIGNMENT No. 2
SEMESTER 1, 2023
1 This group assignment contains four questions. It is worth 20% of the marks for this unit.
2 Each group must email one copy of the assignment to their tutor by 4:30 p.m. Australian Eastern
Standard Time, by Friday, May 5. The assignment must be submitted as a pdf file.
3 Any assignment received after 4:30 p.m. on Friday, May 5 will be deemed late. A penalty of
5% per day will be imposed on assignments submitted after the due date. Any assignment
submitted after 4:30 p.m. Australian Eastern Standard Time on Friday, May 12 will receive a
mark of zero.
3 A special assignment cover sheet will be posted on Moodle. Each member of the group must
type their name on the cover sheet and, if you have the capacity, sign it electronically.
4 Assignments must be either typed or very neatly hand-written. Marks will be deducted for poor
presentation.
5 Please submit one assignment per group.
1
Question 1 (26 marks)
Note that the following notation is used throughout this assignment:
a,b x |a x b.
a,b x |a x b.
a,b x |a x b.
a,b x |a x b.
For example,
10, 12 x |10 x 12.
(a) Let
A 1, 2, 3 and B 2, 4, 6, 8.
i) Is the following relation, R, a function from A to B? Briefly explain.
R 1, 2, 2, 6, 3, 4, 2, 8.
ii) Is the following relation , R, a function from A to B? Briefly explain.
R 1, 6, 2, 6, 3, 2.
(3 marks)
(b) For those relations in (a) which are also functions, specify whether they are surjective, injective
or bijective.
(3 marks)
(c) Let
f : D 0, 1,
where
D x |x 1,
be defined by
fx x 1
x 1
.
i) Use the Theorem 4.1 in the lecture notes to derive
lim
x1
fx.
(2 marks)
ii) Use Theorem 4.1 and Theorem 4.2 to derive
lim
x
fx. (2 marks)
iii) What is fD?
(1 mark)
iv) Is f surjective? Briefly explain.
(1 mark)
2
v) Without using calculus, prove that f is injective.
(4 marks)
vi)
- Does f have an inverse function? Briefly explain.
- If f has an inverse function, derive the inverse function.
(4 marks)
vii) Let
C x 1
4
x 3
4
.
What is f1C, where f1 denotes the inverse function of f.
(2 marks)
(d) Let
a,b .
i) Prove that if
a b
then
|a b| 0.
(1 mark)
ii) Prove that if
|a b| 0,
then
a b. (3 marks)
3
Question 2 (16 marks)
Explicitly reference any theorem(s) to which you appeal when answering this question.
(a) Let
xk
k 2
k2 3
,k .
Use Theorem 3.5 in the lecture notes to derive the limit of xk as k .
(5 marks)
(b) Is xk a bounded sequence? Briefly explain.
(1 mark)
(c) Prove that for all 0 there exists an N such that
k N |xk L| ,
where
L
k
lim k 2
k2 3
.
Report all steps in the proof. No marks will be awarded for an unsupported answer.
(10 marks)
Hints:
i) Suppose that we have an expression of the form
|k 2|.
Note that
|k 2| k 2k 2.
However, when we are considering the limit of a sequence, we are interested in what
happens in the tail of the sequence. Therefore, it is legitimate to assume that
k 2
and assert that
|k 2| k 2k 2.
ii) If
R S,
then if we prove that
k N S , (1)
we have also proved that
k N R . (2)
This is a very useful fact when we want to prove that (2) is true, but it is easier to prove that
(1) is true.
4
Question 3 (16 marks)
Explicitly reference any theorem(s) to which you appeal when answering this question.
(a) Let
xk
1
k
, 1
k
,k
be a sequence of points in 2 and let
y 0, 0.
Derive dxk,y. Report all steps in the derivation. No marks will be awarded for an unsupported
answer
(2 marks)
(b) Use Definition 3.11 in the lecture notes to prove that
k
lim xk
1
k
, 1
k
0, 0.
(4 marks)
(c) Let
x Rn,y Rn, z Rn.
The taxi-cab metric is defined by
dx,y
i1
n
|x i y i|.
i) Prove that
dx,y 0.
(2 marks)
ii) Prove that
dx,y dy,x. (2 marks)
iii) Prove that
dx, z dx,y dy, z. (6 marks)
5
Question 4 (21 marks)
Explicitly reference any theorem(s) to which you appeal when answering this question.
(a) Let
f : D 5, 5
be defined by
fx x2.
Use Definition 4.1 to prove that
Limxx0 fx x0
2.
Hint:
- Factor x2 x02.
- Use the fact that D 5, 5.
(8 marks)
(b) Is f is continuous on D? Briefly explain.
(1 mark)
(c) Let
fx 1
x2 1 x
.
Use Definition 4.3 (a) to prove that
lim
x
fx 0. (7 marks)
(d) Let
gx x2 1 x.
Use the limit laws to compute limx gx.
(5 marks)
6

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