MAT 137
Problem Set #2
Due on Thursday October 10, 2019 by 11:59 pm
Submit via Crowdmark
Instructions
• You will need to submit your solutions electronically. For instructions, see the
MAT137 Crowdmark help page. Make sure you understand how to submit and
that you try the system ahead of time. If you leave it for the last minute and you
run into technical problems, you will be late. There are no extensions for any reason.
• You will need to submit your answer to each question separately.
• You may submit jointly written answers in groups of up to two people. You can work
with anyone in MAT137 from any lecture section. You can work with a different
person for each problem set.
• If you do not jointly write your solutions with someone else then you must submit
your answers individually.
• This problem set is about the definition of a limit and basic properties of limits
(Playlist 2 up to Video 2.13).
Problems
0. Read Notes on Collaboration on the course website. Copy out the following sentence
and sign below it, to certify that you have read the “Notes on Collaboration”.
“I have read and understood the notes on collaboration for this course, as
explained in the course website.”
If submitting as a group of two, both people must sign and submit.
1. (Note: Before you attempt this problem, solve Problem 1 from Practice problems
for Playlist 2 on the course website or Problem 2.1-7 in the textbook. Otherwise you
will find this question too difficult.)
Sketch the graph of a function f that satisfies all 10 conditions below simultaneously.
For this question only, you do not need to prove or explain your answer, as long as
the graph is correct and very clear. If you cannot satisfy all the properties at once,
get as many as you can and state which ones you were unable to achieve.
(a) The domain of f is [−5, 5]
(b) lim
x→a
f(x) exists for every a ∈ (−5, 5),
except a = 0 and a = 3.
(c) f(−2) = 1 and f(3) = −2
(d) lim
x→−2
f(x) = 3
(e) lim
x→0
f(x) 6=∞ or −∞
(f) lim
x→1
f(x) = 0
(g) lim
x→3
f(x) =∞
(h) lim
x→−2
f(f(x)) 6=∞
(i) lim
x→0
[
(f(x))2
]
= 4
(j) lim
x→3
f
( 1
f(x)
)
= −2
2. Let f and g be functions defined at least on an interval centered at a ∈ R, except
possibly at a. Using limit laws, prove that:
IF lim
x→a
f(x) exists and lim
x→a
g(x) exists
THEN, for every integer n ≥ 1, the limit lim
x→a
(
[f(x) + g(x)]n
)
exists.
(Do not use the formal definition of a limit. No epsilons allowed in this question.
Do not use a “power law” for limits. Only use limit laws provided in Video 2.10.)
3. Prove, directly from the formal definition of the limit, that
lim
x→−1
x− 1
x2 − x + 1 = −
2
3
.
Write a proof directly from the definition. Do not use any limit laws.
4. Let f be a function with domain R. Is each of the following claims true or false?
If it is false, show it with a counterexample. If it is true, prove it directly from the
formal definitions of a limit.
(a) IF lim
x→∞
f(x) =∞ THEN lim
x→∞
sin
(
f(x)
)
does not exist.
(b) IF f(−1) = 0 and f(1) = 2 THEN lim
x→∞
f(sin(x)) does not exist.
Note: Before you write this proof, make sure you understand the precise definition
of “the limit is∞” and “the limit does not exist”. Notice that the definition of “the
limit does not exist” is not the negation of “the limit is L”. If your definitions are
not correct, then your proof cannot possibly be correct, and you won’t get any credit.
Make sure to write a formal proof directly from the formal definitions, without using
any limit laws or similar properties.