MATA34 Calculus for Management Assignment 1
Prof. T. Kielstra, Dept. of CMS, University of Toronto, Scarborough
Instructions:
• This assignment is for self-work and is worth 0 percent of your final grade.
• These questions are to give you an opportunity to engage with the material of the course.
• The questions on the quiz will be similar to what you see here. A solid understanding of these questions will help you
start to prepare for the quizzes.
• Each student is different. For some of the topics, you may feel that there are plenty of practice problems, but for others,
there are not enough. When you find that there are not enough problems, I encourage you to look in the notes for
additional suggested exercises.
Limit Questions
Use the graphs to estimate the limits in following questions.
10.1.1
10.1.2
2 y
3
y
2
1
1
x
x
−1
(a) lim f (x)
x→0
10.1.3
1
2
−2 −1
3
(b) lim f (x)
(a) lim f (x)
(c) lim f (x)
x→1
x→−1
x→2
2
x
x→−1
1
2
(b) lim f (x)
x→1
3
10.1.4
(c) lim f (x)
x→2
(a) lim f (x)
x→−1
Use the limit laws from class to solve the following questions.
10.1.9 lim 16
10.1.11 lim (t2 − 5)
10.1.13 lim (3x3 − 4x2 + 2x − 3)
10.1.15 lim
x→2
x→−5
t−2
x→−3 t + 5
x→−2
1
3
(b) lim f (x)
(c) lim f (x)
x→0
−2 −1−1
−2
−3
1
(a) lim f (x)
2
3 y
2
1
3 y
−2 −1
1
x→1
x
1
2
(b) lim f (x)
x→0
3
(c) lim f (x)
x→1
Use the Cancellation law to solve the following limits
x2 + 2x
10.1.21 lim
x→−2 x + 2
√
x2 − x − 2
10.1.23 lim
x→2
x−2
10.1.43 lim
x→6
x−2−2
x−6
Limits at infinity and infinity limits.
Use the graphs to estimate the limits in question 1. Classify each type of discontinuity.
10.2.1
y
(a) lim f (x)
(b) lim f (x)
(c) lim f (x)
(d) lim f (x)
(e) lim− f (x)
(f) lim+ f (x)
x→0
(h) lim− f (x)
(i) lim+ f (x)
(j) lim f (x)
(k) lim f (x)
x→−∞
3
2
1
−5 −4 −3 −2 −1
−1
−2
−3
x→−1
x→−1−
x→0
x
(g) lim f (x)
1 2 3
x→1
x→−1+
x→0
x→1
x→∞
x→1
Compute the limits at infinity for the following.
10.2.9
lim x2
10.2.21 lim
x→−∞
x−5
x→∞ 2x + 1
Compute the infinity limits for each of the following questions.
10.2.9
5x2 + 14x − 3
x→∞
x2 + 3x
lim x2
10.2.21 lim
x→−∞
Compute all the limits for the following question.
10.2.57
(
x
g(x) =
−x
if x < 0
if x > 0
(a) lim+ g(x)
(c) lim g(x)
(b) lim g(x)
(d) lim g(x)
(e)
x→0
x→0
lim g(x)
x→−∞
x→∞
x→0−
Continuity Questions
For each of the following questions, determine if the function is a continuous function. Justify, why or why not.
If the function is not a continuous function, classify all the discontinuities. Question: 11, 12, 29, 34. (Do not
simply analyse the function at the given point.)
10.3.11
10.3.29
(
x+2
f (x) =
x2
10.3.12
if x ≥ 2
if x < 2
10.3.34
(
3
f (x) =
−2
if x ≥ 0
if x < 0
2
1
f (x) = x
0
if x ̸= 0
if x = 0
5
f (x) = x(x − 2)
5x − 1
if x ≥ −1
if x < −1
Definition questions.
Define each of the following terms.
1. The limit of a function at a point.
5. Continuous function
2. An infinite limit.
6. Removable discontinuity.
3. A limit at infinity.
7. Jump discontinuity.
4. Continuous at a point.
8. Infinity discontinuity.
3
0
