Mathematics/MATHS 108
Show all working; problems that do not show their work will typically receive reduced or zero
marks. Late assignments cannot be marked under any circumstances.
1. (Sets.) Let A = (??3; 3);B = [??4; 2];C = [??1; 0) and D = f??4;??5g.
(a) Write the set A B as an interval. Is this an open interval? Is it a closed interval?
(b) Write the set (B n D) [ A as an interval. Is this an open interval? Is it a closed interval?
(c) Write the set B [ C as an interval. Is this an open interval? Is it a closed interval?
2. (Functions and relations.)
(a) Consider the relation
R = f(x; y) : x; y are any two people enrolled in Maths 108 where x is taller than y:g
Is R a function? If it is, explain why, and give its domain; if it is not, explain why not.
(b) Consider the relation
S = f(x; y) : x is any person enrolled in Maths 108, and y is the rst digit of their UoA ID number.g
Is S a function? If it is, explain why, and give its domain; if it is not, explain why not.
3. (Domain and range.) Here are three functions:
(a) f(x) =
x3 ?? 3Ã—2 ?? 4x + 12
x2 ?? 4
; (b) g(x) = ln(16 ?? x2); (c) h(x) = e??x2
:
For each function, describe its natural domain and corresponding range. Explain your reasoning.
4. (Graphing.) Here are two more functions:
(a) f(x) =
8<
:
1=x; x 2 (??1; 0);
sin(x); x 2 [0; );
cos(x); x 2 [;1):
(b) g(x) =
x2 + 1
x2 ?? 2
:
MATHS 108 Page 1 of 2
Draw the graphs of each of these functions by hand. Label any horizontal and vertical asymptotes
and any points of discontinuity. Show the work you used to nd these pieces of information.
5. (Continuity.) Here are two more functions:
(a) f(x) =
cx + 2; x 2 (??1; );
cos(x); x 2 [;1):
(b) h(x) =
x2 ?? 3c2; x 2 (??1; 1];
(cx)2; x 2 (1;1):
For each of these functions, nd a value of c such that the function is continuous everywhere. Explain
how you chose your values of c.
6. (Limits.) Calculate the following four limits, or show that they do not exist. Show your work.
(a) lim
x!0
ln(jx2 ?? 3j) (b) lim
x!1
9Ã—3 ?? 3x
7 ?? 3x + 4Ã—4 (c) lim
x!0
1 ??
p
1 ?? sin2(x)
x
(d) lim
x!1
e2x ?? ex
e2x + ex
7. (Counterexamples.)
The two statements written in the list below are false. For each statement listed below, come
up with a counterexample: that is, nd a function f(x) for statement (a) and a pair of functions
g(x); h(x) for statement (b) that demonstrate why the given statement is false. Explain why your
functions are counterexamples to the given statements.
(a) If a function f(x) is not continuous everywhere (that is, f(x) is a function with at least one
point of discontinuity,) then its square (f(x))2 is also not continuous everywhere.
(b) If the functions g(x) and h(x) do not have a limit as x goes to 0, then their ratio
g(x)
h(x)
also
does not have a limit as x goes to 0.
MATHS 108 Page 2 of 2
Our Service Charter

Excellent Quality / 100% PlagiarismFree
We employ a number of measures to ensure top quality essays. The papers go through a system of quality control prior to delivery. We run plagiarism checks on each paper to ensure that they will be 100% plagiarismfree. So, only clean copies hit customers’ emails. We also never resell the papers completed by our writers. So, once it is checked using a plagiarism checker, the paper will be unique. Speaking of the academic writing standards, we will stick to the assignment brief given by the customer and assign the perfect writer. By saying “the perfect writer” we mean the one having an academic degree in the customer’s study field and positive feedback from other customers. 
Free Revisions
We keep the quality bar of all papers high. But in case you need some extra brilliance to the paper, here’s what to do. First of all, you can choose a top writer. It means that we will assign an expert with a degree in your subject. And secondly, you can rely on our editing services. Our editors will revise your papers, checking whether or not they comply with high standards of academic writing. In addition, editing entails adjusting content if it’s off the topic, adding more sources, refining the language style, and making sure the referencing style is followed. 
Confidentiality / 100% No Disclosure
We make sure that clients’ personal data remains confidential and is not exploited for any purposes beyond those related to our services. We only ask you to provide us with the information that is required to produce the paper according to your writing needs. Please note that the payment info is protected as well. Feel free to refer to the support team for more information about our payment methods. The fact that you used our service is kept secret due to the advanced security standards. So, you can be sure that no one will find out that you got a paper from our writing service. 
Money Back Guarantee
If the writer doesn’t address all the questions on your assignment brief or the delivered paper appears to be off the topic, you can ask for a refund. Or, if it is applicable, you can opt in for free revision within 1430 days, depending on your paper’s length. The revision or refund request should be sent within 14 days after delivery. The customer gets 100% moneyback in case they haven't downloaded the paper. All approved refunds will be returned to the customer’s credit card or Bonus Balance in a form of store credit. Take a note that we will send an extra compensation if the customers goes with a store credit. 
24/7 Customer Support
We have a support team working 24/7 ready to give your issue concerning the order their immediate attention. If you have any questions about the ordering process, communication with the writer, payment options, feel free to join live chat. Be sure to get a fast response. They can also give you the exact price quote, taking into account the timing, desired academic level of the paper, and the number of pages.