MAT 137
Problem Set A
This problem set is intended to help you prepare for Test 1. It is not comprehensive:
it only contains some problems that were not included in past problem sets or in past
tutorials. You do not need to turn in any of these problems.
For further preparation review pre-class quizzes, lecture notes, Problem Set 1, Tutorials 1
through 3, and Unit 1 and 2 practice problems.
Problems
1. In the pre-class video, we have defined a function f is increasing on an interval I if
(i)
Remember that we say a real number x is positive when x > 0 and a real number x
is non-negative when x ≥ 0. Now we can also define a function f is non-decreasing
on an interval I if
(ii)
Now write down a definition of a function f is NOT increasing on an interval I if
(iii)
Below are four claims. Which ones are true and which ones are false? If a claim is
true, prove it. If a claim is false, provide a counterexample and a justification of how
the counterexample shows the claim is false.
(a) If a function f is increasing on I, then it is non-decreasing on I.
(b) If a function f is non-decreasing on I, then it is increasing on I.
(c) If a function f is non-decreasing on I, then it is NOT increasing on I.
(d) If a function f is NOT increasing on I, then it is non-decreasing on I.
2. Let f be a function. Let a, L ∈ R. Assume that f is defined on some open in-
terval around a, except maybe at a. As you know, the definition of the statement
lim f (x) = L is
x→a
, (1) For every ε > 0, there exists δ > 0 such that 0 < |x − a| < δ =⇒
|f (x) − L| < ε.
Below is a list of several statements.
(a) For every ε > 0, there exists δ > 0 such that 0 < |x − a| < δ =⇒ 0 <
|f (x) − L| < ε. Compare this statement with the definition of lim f (x) = L.
x→a
What does this statement mean? Hint: Can you find a function f (x) that
satisfies this statement? Compare f (x) = x and f (x) = x2 sin( x1 ). Use Desmos
to graph these two functions. (https://www.desmos.com/calculator) Do these
two functions satisfy this statement?
(b) For every ε ≥ 0, there exists δ > 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| <
ε. Can you find a function satisfy this statement? If yes, give one example.
If no, can you explain why. Hint: can you write down the negation of this
statement? The negation is true or false?
(c) For every ε > 0, there exists δ > 0 such that |x − a| < δ =⇒ |f (x) − L| < ε.
Compare this statement with the definition of lim f (x) = L. What’s the differ-
x→a
ence? What does this statement
( mean? Hint: sketch the graphs of the functions
x, x 6= 1
f (x) = x and f (x) = . Let a = 1 and L = 1. Check if they are
2, x = 1
satisfies (1) and (c).
(d) For every ε > 0, there exists δ ≥ 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| <
ε. Can you find a function satisfies this statement? If yes, give one example. If
no, can you explain why. Hint: what happens if I take δ = 0?
(e) There exists δ > 0 such that for every ε > 0, 0 < |x−a| < δ =⇒ |f (x)−L| <
ε. What does this statement mean?
3. Let f be a function. Let a, L ∈ R. Assume that f is defined on some open in-
terval around a, except maybe at a. As you know, the definition of the statement
lim f (x) = L is
x→a
(1) For every ε > 0, there exists δ > 0 such that 0 < |x − a| < δ =⇒
|f (x) − L| < ε.
Below is a list of three other statements. Write formal, rigorous proofs for these
statements:
(a) For every ε > 0, there exists δ > 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| ≤
ε.
Prove or disprove that this statement is equivalent to (1).
, Hints: what’s the difference between (1) and (a)? can we say statement (1)
implies this statement (a)? How about the (a) =⇒ (1)? are the epsilons in
this statement and in the definition of lim f (x) = L necessary to be the same?
x→a
(b) Suppose f is defined on some open interval around a and at a.
Prove that if ∀ε > 0, |f (a) − L| < ε, then f (a) = L.
Hint: what is the equivalent statement of “If p, then q.”?
(c) For every δ > 0, there exists ε > 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| <
ε.
What does this statement mean? Hint: let a = 0, L = 1 and f (x) = x. For
every δ > 0, can you find the corresponding ε?
4. Consider the graph of y = f (x) for −9 ≤ x ≤ 9.
y
4
2
−8 −6 −4 −2 2 4 6 8 x
−2
−4
Evaluate each limit. If a limit is ∞ or −∞, clearly state this. If the limit does not
exist and it’s not ∞ or −∞, put “DNE” in the box.
1
(a) lim
x→6 f (x)
(b) lim bf (x)c Here, bxc is the largest integer less or equal to x.
x→−4−
(c) lim f (f (x))
x→−6−
f (x + 5)
(d) lim
x→1 f (x − 1)
Problem Set A
This problem set is intended to help you prepare for Test 1. It is not comprehensive:
it only contains some problems that were not included in past problem sets or in past
tutorials. You do not need to turn in any of these problems.
For further preparation review pre-class quizzes, lecture notes, Problem Set 1, Tutorials 1
through 3, and Unit 1 and 2 practice problems.
Problems
1. In the pre-class video, we have defined a function f is increasing on an interval I if
(i)
Remember that we say a real number x is positive when x > 0 and a real number x
is non-negative when x ≥ 0. Now we can also define a function f is non-decreasing
on an interval I if
(ii)
Now write down a definition of a function f is NOT increasing on an interval I if
(iii)
Below are four claims. Which ones are true and which ones are false? If a claim is
true, prove it. If a claim is false, provide a counterexample and a justification of how
the counterexample shows the claim is false.
(a) If a function f is increasing on I, then it is non-decreasing on I.
(b) If a function f is non-decreasing on I, then it is increasing on I.
(c) If a function f is non-decreasing on I, then it is NOT increasing on I.
(d) If a function f is NOT increasing on I, then it is non-decreasing on I.
2. Let f be a function. Let a, L ∈ R. Assume that f is defined on some open in-
terval around a, except maybe at a. As you know, the definition of the statement
lim f (x) = L is
x→a
, (1) For every ε > 0, there exists δ > 0 such that 0 < |x − a| < δ =⇒
|f (x) − L| < ε.
Below is a list of several statements.
(a) For every ε > 0, there exists δ > 0 such that 0 < |x − a| < δ =⇒ 0 <
|f (x) − L| < ε. Compare this statement with the definition of lim f (x) = L.
x→a
What does this statement mean? Hint: Can you find a function f (x) that
satisfies this statement? Compare f (x) = x and f (x) = x2 sin( x1 ). Use Desmos
to graph these two functions. (https://www.desmos.com/calculator) Do these
two functions satisfy this statement?
(b) For every ε ≥ 0, there exists δ > 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| <
ε. Can you find a function satisfy this statement? If yes, give one example.
If no, can you explain why. Hint: can you write down the negation of this
statement? The negation is true or false?
(c) For every ε > 0, there exists δ > 0 such that |x − a| < δ =⇒ |f (x) − L| < ε.
Compare this statement with the definition of lim f (x) = L. What’s the differ-
x→a
ence? What does this statement
( mean? Hint: sketch the graphs of the functions
x, x 6= 1
f (x) = x and f (x) = . Let a = 1 and L = 1. Check if they are
2, x = 1
satisfies (1) and (c).
(d) For every ε > 0, there exists δ ≥ 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| <
ε. Can you find a function satisfies this statement? If yes, give one example. If
no, can you explain why. Hint: what happens if I take δ = 0?
(e) There exists δ > 0 such that for every ε > 0, 0 < |x−a| < δ =⇒ |f (x)−L| <
ε. What does this statement mean?
3. Let f be a function. Let a, L ∈ R. Assume that f is defined on some open in-
terval around a, except maybe at a. As you know, the definition of the statement
lim f (x) = L is
x→a
(1) For every ε > 0, there exists δ > 0 such that 0 < |x − a| < δ =⇒
|f (x) − L| < ε.
Below is a list of three other statements. Write formal, rigorous proofs for these
statements:
(a) For every ε > 0, there exists δ > 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| ≤
ε.
Prove or disprove that this statement is equivalent to (1).
, Hints: what’s the difference between (1) and (a)? can we say statement (1)
implies this statement (a)? How about the (a) =⇒ (1)? are the epsilons in
this statement and in the definition of lim f (x) = L necessary to be the same?
x→a
(b) Suppose f is defined on some open interval around a and at a.
Prove that if ∀ε > 0, |f (a) − L| < ε, then f (a) = L.
Hint: what is the equivalent statement of “If p, then q.”?
(c) For every δ > 0, there exists ε > 0 such that 0 < |x−a| < δ =⇒ |f (x)−L| <
ε.
What does this statement mean? Hint: let a = 0, L = 1 and f (x) = x. For
every δ > 0, can you find the corresponding ε?
4. Consider the graph of y = f (x) for −9 ≤ x ≤ 9.
y
4
2
−8 −6 −4 −2 2 4 6 8 x
−2
−4
Evaluate each limit. If a limit is ∞ or −∞, clearly state this. If the limit does not
exist and it’s not ∞ or −∞, put “DNE” in the box.
1
(a) lim
x→6 f (x)
(b) lim bf (x)c Here, bxc is the largest integer less or equal to x.
x→−4−
(c) lim f (f (x))
x→−6−
f (x + 5)
(d) lim
x→1 f (x − 1)
