## Mathematics

For Agricultural Engineers

### Practice notes

*By*

*Norbert Bogya, Tam´as Szab´o*

### July 10, 2019

**INVESTING IN YOUR FUTURE**
European Social

Fund

© University of Szeged, Faculty of Science and Informatics, Bolyai Institute

Reviewer: F¨ul¨op Vanda

This teaching material has been made at the University of Szeged, and supported by the European Union. Project identity number: EFOP-3.4.3-16-2016-00014.

**INVESTING IN YOUR FUTURE**
European Social

Fund

Description of the subject 1

1 Set Theory 5

Practice problems . . . 5 Supplementary exercises . . . 8 Quizzes . . . 11

2 Functions 14

Practice problems . . . 14 Supplementery exercises . . . 20 Quizzes . . . 23

3 Limit 27

Practice problems . . . 27 Supplementery exercises . . . 32 Quizzes . . . 35

4 Limits at Infinity, Tangent Line, Derivative 38

Practice problems . . . 38 Supplementary Exercises . . . 43 Quizzes . . . 45

5 Differentiation 48

Practice problems . . . 48 Supplementary exercises . . . 54 Quizzes . . . 56

6 Derivative Tests 59

Practice problems . . . 59 Supplementary exercises . . . 64 Quizzes . . . 68

Midterm #1 72

Version A . . . 73 Version B . . . 77

ii

7 Concavity; Curve Sketching 81

Practice problems . . . 81

Supplementary exercises . . . 86

Quizzes . . . 89

8 Applications of the Derivative 93 Practice problems . . . 93

Supplementary exercises . . . 98

Quizzes . . . 102

9 Sequences; Series 106 Practice problems . . . 106

Supplementary exercises . . . 113

Quizzes . . . 116

10 Integral 119 Practice problems . . . 119

Supplementary Exercises . . . 125

Quizzes . . . 127

11 Matrix Algebra 130 Practice problems . . . 130

Supplementary exercises . . . 140

Quizzes . . . 144

12 Linear Programming 147 Practice problems . . . 147

Supplementary Exercises . . . 155

Quizzes . . . 157

Midterm #2 160 Version A . . . 161

Version B . . . 166

A Preliminaries 170 Exponents and Roots . . . 170

Intervals . . . 171

Algebraic Fractions . . . 171

Linear and Quadratic Equations . . . 171

Linear Functions . . . 172

Quadratic Functions . . . 173

Absolute Value Function . . . 173

Title of the subject: Mathematics for Agricultural Engineers Credits: 3 Type of the subject: compulsory

The ratio of the theoretical and practical character of the subject: 10-90 (credit%) The type of the course: practice

The total number of the contact hours: 2 per week Language: English

Type of the evaluation: 2 written tests

Other methods for evaluation of the student’s competence: 9 quizzes The term of the course: I. semester

Prerequisite of the subject: None

The aim of the subject: The courses are directed to students starting their university stud- ies. It provides an overview of techniques to calculate velocity and acceleration, to estimate the rate of spread of a disease, to set levels of production so as to maximize efficiency, to find the best dimensions of a cylindrical can, to find the age of a prehistoric artifact, and for many other applications. Participants will study Algebra which makes them able to find the values that will maximize or minimize a function subject to restrictions.

Course description:

1. Basic Set Theory 2. Real Valued functions 3. Limit and Continuity 4. Differentiation

5. Sequences and Series 6. Integration

7. Linear Programming

1

Selected bibliography:

• Bogya N. Szab´o T., Mathematics for Agricultural Engineers, SZTE Repository of Educational Resources (this practice note)

• Herbert B. Enderton, Elements of Set Theory, ISBN-13: 978-0122384400, Academic Press, San Diego, 1977

• Saul I. Gass, Linear Programming, ISBN 0-486-43284-X04, Dover Publications, New York, 1958

• George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano: Thomas‘ Cal- culus, ISBN-13: 978-0-321-87896-0, Pearson, 2008

General competence promoted by the subject a) Knowledge

• Familiar with the tools and methods of set theory.

• Know the terminology and computational methods of calculus.

• Understand the differences between formal and informal discussions.

• Understand a new or difficult concept, appreciate its full mathematical precision and outcome.

b) Skills

• Able to interpret and the present the results.

• Able to apply the tools and techniques of mathematics.

• Able to recognize the coherency among the different areas of mathematics.

• Able to participate in mathematical projects under competent supervision.

c) Attitude

• Open to cooperate with her/his classmates.

• Ready to understand the concept of infinity.

• Interested in new results, techniques and methods.

• Aspires to use the abstract termology.

d) Autonomy and responsibility

• Able to solve complex problems independently.

• Provides and requires clear explanation.

• Helps her/his classmates in the completion of their projects.

• Able to create mathematical models consciously.

• Able to do autonomous application of theoritical results.

Special competence promoted by the subject

Knowledge Skills Attitude Autonomy and re-

sponsibilty Know the way to

determine the union and intersection of two or more sets.

Aware of the mean- ing of ”at least”, ”at most” among the real numbers.

Able to carry out the formalization of set operators.

Willing to un- derstand that a paradox is a statement that, despite apparently valid reason- ing from true premises, leads to an apparently-self- contradictory or logically unaccept- able conclusion.

Know the definition of a function, famil- iar with the elemen- tary functions

Able to determine the domain, the ze- roes of a function.

Independently able to draw and explain the linear transform of a function.

Acquire techniques to calculate different types of limits.

By factoring out the common factors able to solve ”0/0”

type limits.

Undestand the dif- ference among the types of limits, and the difference be- tween the limit and continuity.

Know the definition of differentiability, know the derivative of the elementary functions.

Able to calculate the derivative of a composite function.

Independently able to show that a differen- tiable function is con- tinuous.

Know the definition of the sequences and the series.

Able to determine the limit of a serquence, able to calculate the sum of a series.

Ready to explain the difference and the connection be- tween the concept of the sequence and the series.

Recall the definition of the definite and the indefinite integral as well.

Able to calculate the definite and the indefinite integral.

Aspire to solve practical prob- lems.

Discover the im- proper integral independently.

Knowledge Skills Attitude Autonomy and re- sponsibilty

Familiar with the properties of matri- ces. Understand that linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

Able to use the array methods to solve simultaneous equations.

Open to study new techniques and methods.

Can independently give a pictorial repre- sentation of a simple linear program with two variables and six inequalities.

Instructor of the course: Dr. Szab´o Tam´as Zolt´an, associate professor

Teachers: Dr. F¨ul¨op Vanda, senior lecturer; Dr. Szab´o Tam´as, associate professor, Bogya Norbert, assistant research fellow

This practice note was constructed based on the experience and material of the previous years’ Mathematics course for Agricultural Engineers.

There are 12 numbered sections according to the weekly schedule of the practice. In each section there are 3 subsections – Practice problems, Supplementary exercises and Quizzes – in which we explain, deepen and test the material of the actual practice. We start with exercises with well-explained solutions, then we provide several exercises with only final results (without explanations), just for practice. At the end of the section we show some possible quizzes with solutions and scoring, so the students may test their knowledge and score themselves. Moreover, two possible midterms are included among the homework sections to prepare them for the exam.

We are sure that if students attend the classes and use this note, then they will be able to characterise mathematical problems, memorise the types of exercises, recall the possible methods to solve and use them. With their improved mathematical skills, they will be able to explain their own solution clearly and think in a more abstract way.

*The Authors*

## Set Theory

### Practice problems

### Elements, subset, power set

Exercise 1. Give the elements of the following sets.

(a) *A*=*{x*: *x*is positive integer and *x*^{2} *<*20*}*
(b) *B* =*{*1*−n*^{2} *|n* is even and 3*≤n≤*6*}*

(c) The set*C* of even prime numbers greater than 10.

Solution.

(a) Positive integers are 1,2,3,4, . . . and such a number also has to satisfy the following
property: its square should be smaller then 20. It is correct for only the numbers 1,2,3,4,
so *A*=*{*1,2,3,4*}.*

(b) There are two restrictions to*n: it has to be even and between 3 and 6. Only the numbers*
4 and 6 (equality is allowed) satisfy these conditions, hence *B* = *{*1*−* 4^{2}*,*1*−*6^{2}*}* =
*{−*15,*−*35*}*.

(c) Set *C* is given by a description. We look for the even primes greater than 10. There is
only one even prime number, that is 2. But 2 is not greater then 10. This set has no
element, so it is*the* empty set,*C* =*∅*.

Exercise 2. If*a, b, c, d*are distinct objects, determine which of the five sets*{a, b, c}*,*{b, c, a, b}*,
*{c, a, c, b}*, *{b, c, b, a}*, *{a, b, c, d}* are equal.

Solution. A set is irrelevant to multiplicity and ordering. So we can ignore the multiple appearances of an object and their order:

*{a, b, c}*=*{b, c, a, b}*=*{c, a, c, b}*=*{b, c, b, a}.*

5

Exercise 3. Let*A*=*{a, b, c}*,*B* =*{a, b}*,*C* =*{a, b, d}*, *D*=*{a}*and *E* =*{b, c}*, where *a, b, c*
are distinct objects. State whether each of the following statements is true or false:

(a) *B* *⊆A,* (b) *E* =*C,* (c) *D⊆B,* (d) *D⊆A,* (e) *A*=*B.*

Solution. Notation *A⊆B* means that each element of set *A* is also a member of set *B.*

(a) True, because*B* has two elements: *a, b* and they are elements of *A, too.*

(b) True, because*E* contains *c, butC* does not, so their elements are not the same.

(c) False, since the only element of *D* is*a* and it is contained in *B* as well.

(d) True, the only element of *D* is*a* and it is an element of*A, too.*

(e) False, since*c*is contained in only one of the sets.

Exercise 4. Which of the following sets is *the* empty set?

(a) *H* =*{x*: *x* is an odd integer and *x*^{2} = 4*}*
(b) *K* =*{x*: *x*is an integer and *x*+ 8 = 8*}*

(c) *L*=*{x*: *x* is a positive integer and *x <*1*}*
Solution.

(a) There are only two numbers whose square is 4, they are the number 2 and*−*2. But they
are even, not odd, so*H* does not contain any element, *H* =*∅*.

(b) The equation of the restriction has exactly one solution which is 0. Zero is an integer,
hence *K* =*{*0*} *=*∅*.

(c) The smallest positive integer is 1, but it is not smaller then 1, as the restriction requires.

Hence, *L*=*∅.*

Exercise 5. Give the elements of the following power sets.

(a) *P*(*{*1,2*}*) (b) *P*(*{a,*1,5*}*) (c) *P*(*∅*) (d) *P*(*P*(*∅*))

Solution. Power set of a set *A* is the set of all subsets of *A. So we have to determine the*
subsets of the given sets. Empty set is always contained in any power set.

(a) Denote*{*1,2*}*with*A. The empty set is a subset of any set, and that is the only subset with*
no elements. The subsets of*A*with only one element are*{*1*}*and*{*2*}*. Furthermore,*A*has
only one subset with two elements, that is the whole *A. Hence,* *P*(A) =*{∅,{*1*},{*2*}, A}.*
(b) Let *B* =*{a,*1,5*}*. Again, the empty set is the only subset of *B* with zero elements. The
subsets of *B* with one element are *{a}*, *{*1*}* and *{*5*}*. But now, there are several subsets
with two elements: *{a,*1*}*, *{a,*5*}* and *{*1,5*}*. Of course, the whole set *B* is a subset of
itself. We have *P*(B) = *{∅,{a},{*1*},{*5*},{a,*1*},{a,*5*},{*1,5*}, B}.*

(c) The empty set has no element, so it has only one subset, itself: *P*(*∅*) =*{∅}.*

(d) From the previous part we know that *P*(*∅*) =*{∅}.* This set has one element, so it has a
subset with one element, that is *{∅}*. Hence,*P*(*P*(*∅*)) =*{∅,{∅}}.*

### Operations between sets

Exercise 6. Let *A* = *{*1,2,3,4,5*}* and *B* = *{*2,4,6,8,9*}* be two subsets of the universal set
*U* =*{*1,2, . . . ,10*}*. Give the elements of the sets*A∪B*,*A∩B*,*A\B*,*B* *\A,A* and *AB.*

Solution. We illustrate the sets with a Venn diagram.

1 3 5 6 8 9

7 10

*A* *B*

2 4

• Intersection means the common part of the sets, so *A∩B* =*{*2,4*}*.

• In the union we collect all elements from all of the sets: *A∪B* =*{*1,2,3,4,5,6,8,9*}.*

• Set *A\B* contains objects that are in *A* but not in *B, hence* *A\B* =*{*1,3,5*}*. Similarly,
*B\A*=*{*6,8,9*}*.

• The complement of a set contains elements that are not in the set: *A*=*{*6,8,9,7,10*}*.

• Symmetric difference of *A* and *B* is the set of elements, which occur exactly once in the
sets *A* and *B*, so *AB* =*{*1,3,5,6,8,9*}*.

Exercise 7. Illustrate the following sets on the number line.

(a) [0,2]*∩*(1,3) (b) [0,3)*∪*(*−*1,2] (c) (0,5]*\*[1,2]

Solution.

(a) We sketch the intervals with colour blue, their common part is coloured by red. Number 1 is not contained in the intersection, because it is not in the interval (1,3). But number 2 is an element of both of the sets, so it is in the intersection as well.

0 2

1 3

1 2

(b) Now we have to calculate a union. We collect the points of the axis that are covered by at least one of the blue intervals.

0 3

*−*1 2

*−*1 3

(c) We delete the common parts from the first set. We have to care about the solid and empty circles: [1,2] is a closed interval, so we delete its endpoints (as well) from (0,5].

0 5

1 2

0 1 2 5

### Supplementary exercises

### Elements, subset, power set

Exercise 1. Let *A* = *{∅,{∅},{∅,{∅}}}*. Determine whether each of the following statements
is true or false.

(a) *∅ ∈A*
(b) *∅ ⊆A*

(c) *{∅} ∈A*
(d) *{∅} ⊆A*

(e) *{{∅}} ∈A*
(f ) *{{∅}} ⊆A*

(g) *{∅,{∅}} ∈A*
(h) *{∅,{∅}} ⊆A*
Solution.

(a) True (b) True

(c) True (d) True

(e) False (f ) True

(g) True (h) True Exercise 2. Determine whether each of the following statements is true or false.

(a) *a∈ {*1, a, b,2*}*
(b) *a⊆ {*1, a, b,2*}*

(c) *{a} ∈ {*1, a, b,2*}*
(d) *{a} ⊆ {*1, a, b,2*}*

(e) *{a,*2*} ∈ {*1, a, b,2*}*
(f ) *{a,*2*} ⊆ {*1, a, b,2*}*
(g) *{a,*2,3*} ∈ {*1, a, b,2*}*
(h) *{a,*2,3*} ⊆ {*1, a, b,2*}*
Solution.

(a) True (b) False (c) False (d) True

(e) False
(f ) True
(g) False
(h) False
Exercise 3. Give the elements of the set *P*(*P*(*P*(*∅*))).

Solution. *P*(*P*(*P*(*∅*))) =*{∅,{∅},{{∅}},{∅,{∅}}}*

Exercise 4. Give the power sets of the following sets.

*A*=*{*5*}* *B* =*{*2, a*}* *C* =*{*1,0,*∅}* *D*=*{*1,2, c, d*}*
Solution.

*P*(A) =

*∅,{*5*}*
*P*(B) =

*∅,{*2*},{a},{*2, a*}*
*P*(C) =

*∅,{*1*},{*0*},{∅},{*1,0*},{*1,*∅},{*0,*∅},{*1,0,*∅}*

*P*(D) =

*∅,{*1*},{*2*},{c},{d},{*1,2*},{*1, c*},{*1, d*},{*2, c*},*

*{*2, d*},{c, d},{*1,2, c*},{*1,2, d*},{*1, c, d*},{*2, c, d*},{*1,2, c, d*}*

### Operations between sets

Exercise 5. Let*A* =*{a, b, c, d}*, *B* =*{b, d, f, h}*,*C* =*{c, d, e, f}*. Find
(a) *A∩B,* *A∩C,* *B∩C,*

(b) *A∪B,* *A∪C,* *B∪C,*

(c) *A\B,* *B\A,* *A\C,* *C\A,C\B,* *B\C.*

Solution.

(a) *A∩B* =*{b, d}*, *A∩C* =*{c, d}*, *B∩C* =*{d, f}*

(b) *A∪B* =*{a, b, c, d, f, h}*, *A∪C* =*{a, b, c, d, e, f}*,*B* *∪C*=*{b, c, d, e, f, h}*

(c) *A\B* =*{a, c}*,*B\A*=*{f, h}*,*A\C* =*{a, b}*,*C\A* =*{e, f}*,*C\B* =*{c, e}*,*B\C* =*{b, h}*
Exercise 6. Let Rbe the set of real numbers, *A*=*{x∈*R: 1*≤x≤*3*}*and *B* =*{x∈*R: 2*≤*
*x≤*4*}*. Find the following sets.

(a) *A∪B*
(b) *A∩B*

(c) (R*\A)∩B*
(d) (R*\B*)*∩A*

(e) (R*\A)∩*(R*\B)*

(f) (R*\B)∩*(R*\A)*
(g) (R*\A)∪*(R*\B)*
(h) *B* *∪*[A*∩*(R*\B*)]

(i) [(R*\A)∩B]∪*[(R*\B)∩A]*

Solution.

(a) *{x∈R*: 1*≤x≤*4*}*
(b) *{x∈R*: 2*≤x≤*3*}*
(c) *{x∈R*: 3*< x≤*4*}*
(d) *{x∈R*: 1*≤x <*2*}*

(e) *{x∈R*: *x <*1 or 4*< x}*

(f) *{x∈R*: *x <*1 or 4*< x}*
(g) *{x∈R*: *x <*2 or 3*< x}*
(h) *{x∈R*: 1*≤x≤*4*}*,

(i) *{x∈R*: 1*≤x <*2 or 3*< x≤*4*}*

Exercise 7. Let *A* =*{x* *∈* Z: *x* is a multiple of 10*}* and *B* = *{x* *∈* Z: *x*is a multiple of 15*}*.
What is *A∩B? Can you generalize this result?*

Solution. *A∩B* =*{x∈*Z: *x* is a multiple of 30*}.*

In general, if *A* = *{x* *∈*Z: *x* is a multiple of *m}* and *B* =*{x* *∈* Z: *x* is a multiple of *n}*, then
*A∩B* =*{x∈*Z: *x*is a multiple of *k}* where *k* is the least common multiple of *m* and *n.*

Exercise 8. Let *A* = *{*1,2,3,8,9*}*, *B* = *{*2,4,6,8*}*, *C* = *{*3,6,9*}*. Determine *A\B,* *C\A,*
(A*\B)∩C, (B* *∪C)\*(A*\C),* *P*(C), *P*(A*\B).*

Solution.

• *A\B* =*{*1,3,9*}*

• *C\A*=*{*6*}*

• (A*\B)∩C* =*{*3,9*}*

• (B*∪C)\*(A*\C) =* *{*3,4,6,9*}*

• *P*(C) =*{∅,{*3*},{*6*},{*9*},{*3,6*},{*3,9*},{*6,9*}, C}*

• *P*(A*\B) =* *{∅,{*1*},{*3*},{*9*},{*1,3*},{*1,9*},{*3,9*},{*1,3,9*}}*

Exercise 9. Let *U* = *{a, b, c, d, e}* be the universal set, *A* = *{a, b, c, d}*, *B* = *{d, e}* and
*C* =*{a, b, e}*. Determine the elements of the following sets:

*A∪B,* *A∩B,* *B,* *A\B,* *AB,* (A*C)\B.*

Solution.

• *A∪B* =*U*

• *A∩B* =*{d}*

• *B* =*{a, b, c}*

• *A\B* =*{a, b, c}*

• *AB* =*{a, b, c, e}*

• (A*C)\B* =*∅*

Exercise 10. Let*A* =*P*(*{a, b}*) and *B* =*P*(*{b, c}*). Determine the elements of the following
sets:

*A∪B,* *A∩B,* *A\B,* *B\A,* *AB.*

Solution.

• *A∪B* =*{∅,{a},{b},{a, b},{c},{b, c}}*

• *A∩B* =*{∅,{b}}*

• *A\B* =*{{a},{a, b}}*

• *B\A*=*{{c},{b, c}}*

• *AB* =*{{a},{c},{a, b},{b, c}}*

Exercise 11. Sketch the following sets: (*−*2; 3]*\*[1; 4], (*−*2; 3]*∪*[1; 4], (*−*2; 3]*∩*[1; 4].

Solution.

*−*2 3

1 4

*−*2 1

*−*2 4

1 3

(*−*2; 3]*\*[1; 4]

(*−*2; 3]*∪*[1; 4]

(*−*2; 3]*∩*[1; 4]

Exercise 12. Sketch the following sets: (2,6]*\*(1; 4), (2,6]*∩*(1; 4), (2,6]*∪*(1; 4).

Solution.

2 6

1 4

4 6

2 4

1 6

(2,6]*\*(1; 4)
(2,6]*∩*(1; 4)
(2,6]*∪*(1; 4)

### Quizzes

*Version A*

Exercise 1. Let*A*=*{*1,2,3,8,9*}*, *B* =*{*2,4,6,8*}* and*C* =*{*3,6,9*}*. Give the elements of the
following sets.

(a) *AB*

(b) (A*∩B)∪C*
(c) (C*∪B)\*(A*∩C)*
(d) *P*(C)

Exercise 2. Sketch the following sets on the number line.

(a) (*−*1; 3]*∩*[1; 2]

(b) (*−*2; 2)*∪*[*−*1; 4]

(c) (*−*2; 3]*\*(0; 5)
*Version B*

Exercise 1. Calculate the power set of (A*∪B)∩*(B*C), whereA*=*{*1,2,4,7*}*,*B* =*{*2,5,6,8*}*,
*C* =*{*3,5,6,8*}* and the universal set is *U* =*{*1,2,3,4,5,6,7,8*}*.

Exercise 2. Give the following sets as intervals.

(a) *{x*: *x∈*R*, x*^{2} *<*1*}*

(b) *{x*: *x∈*R*, x≤*2*} ∩*(*−*3,5)
(c) [*−*3,4)*\*[1,5)

### Solution of Quizzes

*Version A*

Solution of Exercise 1.

(a) *AB* =*{*1,3,4,9,6*}*

(b) (A*∩B)∪C* =*{*2,8*} ∪ {*3,6,9*}*=*{*2,3,6,8,9*}*

(c) (C*∪B)\*(A*∩C) =* *{*2,3,4,6,8,9*} \ {*3,9*}*=*{*2,4,6,8*}*
(d) *P*(C) =*{∅,{*3*},{*6*},{*9*},{*3,6*},{*3,9*},{*6,9*},{*3,6,9*}}*

Solution of Exercise 2.

(a) (*−*1; 3]*∩*[1; 2]

*−*1 3

1 2

1 2

(b) (*−*2; 2)*∪*[*−*1; 4]

*−*2 2

*−*1 4

*−*2 4

(c) (*−*2; 3]*\*(0; 5)

*−*2 3

0 5

*−*2 0

1 pt

2 pt

3 pt

4 pt

5 pt

*Version B*

Solution of Exercise 1.

• *B* =*{*1,3,4,7*}*

• *A∪B* =*{*1,2,3,4,7*}*

• *BC* =*{*2,3*}*

• (A*∪B)∩*(B*C) ={*1,2,3,4,7*} ∩ {*2,3*}*=*{*2,3*}*

• *P*(*{*2,3*}*) = *{∅,{*2*},{*3*},{*2,3*}}*

Solution of Exercise 2.

(a)

*x*^{2} *<*1 *⇐⇒ |x|<*1 *⇐⇒ −*1*< x <*1
*{x*:*x∈*R*, x*^{2} *<*1*}*= (*−*1,1)

(b) We have to delete the numbers greater than 2 from the interval (*−*3,5). The remaining
interval is (*−*3,2].

(c)

[*−*3,4)*\*[1,5) = [*−*3,1)

*−*3 4

1 5

*−*3 1

1 pt

2 pt

3 pt

4 pt

5 pt

## Functions

### Practice problems

### Functions

Exercise 1. If *y* = *f(x) = 2x*^{3} *−*3x^{2} + 1, compute the following values and plot the corre-
sponding points in the coordinate system.

(a) *f*(0) (b) *f*(1) (c) *f(2)* (d) *f*(*−*1)

Solution. Substituting the given numbers for *x* into the function we get the *y-coordinate of*
the corresponding points.

(a) *f(0) = 2(0)*^{3}*−*3(0)^{2}+ 1 = 1

Thus *y* = 1 when *x* = 0, it gives the
ordered pair*A(0,*1).

(b) *f*(1) = 2(1)^{3}*−*3(1)^{2}+ 1 = 0

Thus *y* = 0 when *x* = 1, it gives the
ordered pair*B*(1,0).

(c) *f*(2) = 2(2)^{3}*−*3(2)^{2}+ 1 = 5

Thus *y* = 5 when *x* = 2, it gives the
ordered pair*C(2,*5).

(d) *f*(*−*1) = 2(*−*1)^{3}*−*3(*−*1)^{2}+ 1 =*−*4
Thus *y* =*−*4 when *x* =*−*1, it gives the
ordered pair*D(−*1,*−*4).

*−*1 1 2

*−*4
1
5

*A*
*B*

*C*

*D*

Exercise 2. Find the following expressions, if *g(x) = 3x−*2.

(a) *g(2a−*1) (b) *g(x*+*h)* (c) *g(x*+*h)−g(x)*

*h* *.*

14

Solution. We substitute the given expressions into the function. After that we make some simplifications.

(a)

*g(2a−*1) = 3(2a*−*1)*−*2 = 6a*−*3*−*2 = 6a*−*5
(b)

*g(x*+*h) = 3(x*+*h)−*2 = 3x+ 3h*−*2
(c)

*g(x*+*h)−g(x)*

*h* = (3x+ 3h*−*2)*−*(3x*−*2)

*h* = 3x+ 3h*−*2*−*3x+ 2

*h* = 3h

*h* = 3
Exercise 3. Identify the inner and outer parts of the following composite functions.

(a) *√*^{3}

1*−x* (b) *e*^{−x}^{2} (c) log_{5}(2*−x*^{3}) (d) (2*−*10* ^{x}*)

^{3}Solution. The outer function corresponds to the last operation we take, calculating a specific value of the given function.

(a) (b) (c) (d)

Outer function: *√*^{3}

*x* *e** ^{x}* log

_{5}

*x*

*x*

^{3}Inner function: 1

*−x*

*−x*

^{2}2

*−x*

^{3}2

*−*10

^{x}Exercise 4. Let*f*(x) = 2x*−*3 and *g(x) =x*^{2}+ 1. Find the following expressions.

(a) *f*(g(x)) (b) *g(f(x))*

Solution. We make composite functions.

(a)

*f(g(x)) = 2(g(x))−*3 = 2(x^{2}+ 1)*−*3 = 2x^{2}*−*11
(b)

*g(f*(x)) = (f(x))^{2}+ 1 = (2x*−*3)^{2}+ 1 = 4x^{2}*−*12x+ 10

Exercise 5. Use the vertical line test to determine whether the given relation is a function by inspecting its graph.

(a) (b) (c)

Solution.

(a) (b) (c)

Yes, it passes the vertical line test.

Yes, it passes the vertical line test.

No, it fails the vertical line test.

Exercise 6. Find the domain of the following functions.

(a) *y*= 2x^{3} (b) *y* =*√*

3*−*2x (c) *y*= 7*−*1/x^{2} (d) *y*= ln(5x+ 4)
Solution. The domain of a function is assumed to be the largest subset of the real numbers
for which the given formula produces a real number.

(a) There are no restrictions on the numbers substituted for*x, so the domain consists of all*
real numbers, *x∈*R.

(b) Since *√*

3*−*2x is not definied for 3*−*2x <0 (that is for *x >*3/2), we conclude that the
domain of the function is the interval *x≤*3/2 or (*−∞,*3/2].

(c) Since a denominator of a fraction cannot be zero, the domain consists of all real numbers
except the zero point of *x*^{2}, that is 0.

(d) The argument of a logarithm is always positive, so the domain is the set of all real numbers
satisfying 5x+ 4*>*0, that means *x >−*4/5.

Exercise 7. Draw the graphs of the following functions.

(a) *f*(x) =

*−x,* *x <*0

*−x*^{2}*,* 0*≤x <*2
*x−*1, 2*≤x*

(b) *f(x) =*

2, *x≤*1/2

1

*x**,* 1/2*< x <*3

*−*3, 3*≤x*
Solution.

(a) (b)

Exercise 8. Give the equation of the straight lines which correspond the following descriptions.

(a) The line goes through the point*P*(2,*−*1) and its slope is *m*= 1/2.

(b) The line goes through the points*A(−*1,2) and*B*(*−*2,5).

Solution.

(a) We use the equation*y* =*mx*+*b* of a straight line, *m*is given. So the half-ready equation
is

*y*= 1
2*x*+*b.*

After substitution of the given point we get a value for*b.*

*y* = 1

2 *·x*+*b*

*−*1 = 1

2 *·*2 +*b*

*−*1 = 1 +*b*

*−*2 = *b*
So the equation of the line is*y* = 1

2*x*+*b.*

(b) From two points of a line, we can calculate its slope:

*m* = ∆y

∆x = *y*2*−y*1

*x*2*−x*1

= 5*−*2

*−*2*−*(*−*1) =*−*3.

Then we can follow the method used in the exercise above. We have the slope and a given
point of the line. After substituting the point*B* into the half-ready equation*y*=*−*3x+*b,*
we get *b* =*−*1. The final equation of the line is*y* =*−*3x*−*1.

### Manipulating Graphs

Exercise 9. Match the equations listed to the graphs in the accompanying figure.

1. The base function is *y*=*x*^{2}.
(a) *y*= (x*−*1)^{2}*−*4

(b) *y*= (x*−*2)^{2}+ 2
(c) *y*= (x+ 2)^{2} + 2
(d) *y*= (x+ 3)^{2} *−*2

*−*3*−*2 1 2

*−*4

*−*2
2

2. The base function is *y*= ln*x.*

(a) *y*= ln 3x
(b) *y*= ln^{1}_{2}*x*
(c) *y*= 3 ln*x*
(d) *y*= ^{1}_{2}ln*x*

3. The base function is *y*=*√*
*x.*

(a) *y*=*√*

*−x*
(b) *y*=*−√*

*x*

Solution.

1. The graph of the base function*y*=*x*^{2}
(a) moves right 1 unit and moves down 4

units, so*y*= (x*−*1)^{2}*−*4 is red;

(b) moves right 2 units and moves up 2
units, so*y*= (x*−*2)^{2}+ 2 is orange;

(c) moves left 2 units and moves up 2
units, so *y*= (x+ 2)^{2}+ 2 is light blue;

(d) moves left 3 units and moves down 2
units, so*y*= (x+ 3)^{2}*−*2 is blue.

*−*3*−*2 1 2

*−*4

*−*2
2

2. The graph of the base function*y*= ln*x*
(a) contracts in the horizontal direction, so

*y*= ln 3x isred;

(b) stretches in the horizontal direction, so
*y*= ln^{1}_{2}*x* isblue;

(c) stretches in the vertical direction, so
*y*= 3 ln*x* islight blue;

(d) contracts in the vertical direction, so
*y*= ^{1}_{2}ln*x* isorange.

3. The graph of the base function*y*=*√*
*x*
(a) is reflected through the *y-axis, so*

*y*=*√*

*−x* is red;

(b) is reflected through the *x-axis, so*
*y*=*−√*

*x* is blue.

Exercise 10. Graph each function, not by plotting points, but by starting with the graph of
the function *f*0(x) =*x*^{3} and applying the appropriate transformations.

(a) *y*= (x*−*1)^{3}*−*2 (b) *y*= (1*−x)*^{3}+ 2
Solution.

(a) 1. step:

*f*_{1}(x)=*f*_{0}(x*−*1) = (x*−*1)^{3}
Graph of*f*0 moves right 1 unit.

2. step:

*f*_{2}(x)=*f*_{1}(x)*−*2 = (x*−*1)^{3}*−*2

Graph of*f*1 moves down 2 units. ^{1}

(b) 1. step:

*f*1(x)=*f*0(x+ 1) = (x+ 1)^{3}
Graph of*f*0 moves left 1 unit.

2. step:

*f*2(x)=*f*1(*−x)*= (*−x*+ 1)^{3}

Graph of*f*1 is reflected to the *y-axis.*

3. step:

*f*3(x)=*f*2(x)+ 2 = (1*−x)*^{3}+ 2
Graph of*f*2 moves up 2 units.

*−*1 1

### Supplementery exercises

### Functions

Exercise 1. Substitute the following expressions into the function*f(x) = 1−*log_{2}(x^{2}*−*3) and
give their simplest form.

(a) *x*=*−*2 (b) *x*=*√*

11 (c) *x*=*√*

*t*
Solution.

(a) *f*(*−*2) = 1 (b) *f(√*

11) =*−*2 (c) *f(√*

*t) = 1−*log_{2}(t*−*3)

Exercise 2. Substitute the following expressions into the function *g(x) =* *x*^{2}
2^{x}*·√*

*x* and give
their simplest form.

(a) *x*= 1 (b) *x*= 4 (c) *x*=*t*^{2}

Solution.

(a) *g(1) =* 1

2 (b) *g(4) =* 1

2 (c) *g(t*^{2}) = *t*^{3}

2^{t}^{2}
Exercise 3. Give the *f*(g(x)) and*g(f*(x)) composite functions, where

(a) *f*(x) = *√*^{3}

*x,* *g(x) = 2−x*^{2};
(b) *f(x) = 2x−*3, *g(x) =e** ^{x}*;

(c) *f(x) = 1−√*

*x,* *g(x) = log*_{3}*x;*

(d) *f(x) =* *x*^{2}*−*1, *g(x) = 3/x.*

Solution.

(a) *f*(g(x)) = *√*^{3}

2*−x*^{2} *g(f*(x)) = 2*−√*3

2*−x*^{2}2

(b) *f(g(x)) = 2e*^{x}*−*3 *g(f(x)) =e*^{2x−3}
(c) *f(g(x)) = 1−*

log_{3}*x* *g(f*(x)) = log_{3}(1*−√*
*x)*
(d) *f(g(x)) =* 9

*x*^{2} *−*1 *g(f*(x)) = 3
*x*^{2}*−*1

Exercise 4. Determine the domain of the following functions.

(a) *f*(x) = 4(x*−*3)^{2}
(b) *f(x) = log*_{2}(1*−*3x)

(c) *f*(x) = log_{2}(1*−√*
*x)*
(d) *f(x) =* 3

4*−x*^{2}

(e) *f(x) =* *√*^{3}
3 + 2x
(f) *f(x) =* *√*

*x−*log_{2}(x*−*3)
(g) *f(x) =* *√*

9*−x*^{2}
(h) *f(x) =* 2

*e*^{1+x}

Solution.

(a) *x∈*R
(b) *x <*1/3

(c) *x∈*[0,1)
(d) *x∈*R*\ {−*2,2*}*

(e) *x∈*R
(f) *x >*3

(g) *x∈*[*−*3,3]

(h) *x∈*R
Exercise 5. Draw the graph of the following functions.

(a) *f(x) =*

*x*^{2}*,* if *x <*0

*−x,* if 0*≤x≤*1
2^{x}*,* if *x >*1

(b) *g(x) =*

*x*^{4}*−*1, if *x <*0

*√x,* if 0*≤x <*1
1 + _{x}^{1}*,* if *x≥*1

Solution. (a) (b)

Exercise 6. Give the equation of the straight lines which satisfy the following properties.

(a) The line goes through the point*P*(0,3) and it has a slope *m*= 2.

(b) The line goes through the points*A(−*1,5) and*B*(2,2).

(c) The line goes through the points*A(−*1,1) and*B*(0,*−*1).

Solution.

(a) *y*= 2x+ 3 (b) *y*=*−x*+ 4 (c) *y*=*−*2x*−*1

### Manipulating Graphs

Exercise 7. Using transformation steps draw the graphs of the following functions.

(a) *f*(x) = (x*−*2)^{3}+ 1
(b) *f*(x) = 2*√*^{3}

*x−*1
(c) *f*(x) = 1*−*1

2log_{2}*x*
(d) *f(x) = log*_{2}(3*−x)*

(e) *f(x) = 2*^{1}^{−}* ^{x}*
(f)

*f(x) =*

1*−* *x*
2

2

(g) *f(x) =* *√*
2x*−*3
(h) *f(x) =* *x*

*x*+ 1

Solution.

(a)

2 1

(b)

*−*1

(c) (d)

1 2

(e) (f)

2

(g)

3/2

(h) *x*

*x*+ 1 = *x*+ 1*−*1

*x*+ 1 = 1*−* 1
*x*+ 1

### Quizzes

*Version A*

Exercise 1. Substitute the following expressions into the function *g(x) = log*_{3}(2x*−*5) and
give their simplest form.

(a) *x*= 3 (b) *x*= 7 (c) *x*= 1

Exercise 2. Draw the graph of the function

*f(x) =*

*−*2x*−*1, if *x <*0,

*√x,* if 0*≤x≤*1,
1, if *x >*1.

Exercise 3. Using transformation steps draw the graph of the function*f*(x) =*−√*

2x+ 4.

*Version B*

Exercise 1. Give the domain of the function*f(x) =* *e*^{1}^{−}^{1/x}*−*log_{2}(2x+6) as unions of intervals.

Exercise 2. Give the equation of the straight line that passes through the points (2,*−*1) and
(*−*2,4).

Exercise 3. Using transformation steps draw the graph of the function*f*(x) = *√*^{3}

4*−*2x.

*Version C*

Exercise 1. Let *f(x) =√*

1*−x,* *g(x) = 1−*2x. Determine the functions*f*(g(x)) and *g(f(x))*
and calculate the values *f(g(2)) and* *g(f(2)).*

Exercise 2. Give the equation of the line which goes through the point*P*(1,*−*2) and it has a
slope *m*=*−*^{2}_{3}.

Exercise 3. Use linear transformation steps to sketch the graph of the function
*f(x) = 1−* 2

*x−*1*.*

### Solution of Quizzes

*Version A*

Solution of Exercise 1.

(a) *g(3) = log*_{3}(2*·*3*−*5) = log_{3}1 = 0

(b) *g(7) = log*_{3}(2*·*7*−*5) = log_{3}9 = log_{3}3^{2} = 3
(c) The function is not defined at *x*= 1.

Solution of Exercise 2.

*−*1/2 1

*−*1

Solution of Exercise 3.

• *f*0(x) = *√*
*x*

• *f*0(x+ 4) =*√*

*x*+ 4 =*f*1(x)
Graph of *f*0 moves left 4 units.

• *f*1(2x) = *√*

2x+ 4 =*f*2(x)

Graph of *f*1 is compressed horizontally.

• *−f*2(x) = *−√*

2x+ 4 =*f(x)*

Graph of *f*_{2} is reflected through the *x-*
axis.

*−*4 *−*2 2

*−*2
2

1 pt

2 pt

3 pt

4 pt

5 pt

*Version B*

Solution of Exercise 1. We have two restrictions: *x* = 0 and 2x+ 6 *>* 0. The domain
consists of numbers greater than *−*3 except 0, that is (*−*3,0)*∪*(0,*∞*).

Solution of Exercise 2.

The slope of the line is *m*= 4*−*(*−*1)

*−*2*−*2 =*−*5
4.
Using *y* = *mx*+*b, we have* *y* = *−*5

4*x*+*b. Substituting (2,−*1), we have *−*1 = *−*5

4 *·*2 +*b,*
hence *b*= 3

2.

The equation of the line is *y*=*−*5
4*x*+ 3

2. Solution of Exercise 3.

• *f*0(x) = *√*^{3}
*x*

• *f*0(x+ 4) = *√*^{3}

*x*+ 4 =*f*1(x)
Graph of *f*0 moves left 4 units.

• *f*1(2x) = *√*^{3}

2x+ 4 =*f*2(x)

Graph of *f*1 is compressed horizontally.

• *f*2(*−x) =* *√*^{3}

*−*2x+ 4 =*f*(x)

Graph of *f*2 is reflected through the *y-*
axis.

*−*4 *−*2 1
1

1 pt

2 pt

3 pt

4 pt

5 pt

*Version C*

Solution of Exercise 1.

• *f*(g(x)) =

1*−*(1*−*2x) = *√*
2x

• *g(f(x)) = 1−*2(*√*

1*−x) = 1−*2*√*
1*−x*

• *f*(g(2)) =*√*

2*·*2 =*√*
4 = 2

• *g(f(x)) is not defined at* *x*= 2

Solution of Exercise 2. By *y* = *mx*+*b, we have* *y* = *−*2

3*x*+*b. Substituting* *P* we get

*−*2 =*−*2

3 +*b, thusb* =*−*4

3. The equation of the line is*y*=*−*2
3*x−* 4

3. Solution of Exercise 3.

• *f*0(x) = 1
*x*

• *f*0(x*−*1) = 1

*x−*1 =*f*1(x)
Graph of *f*0 moves right 1 unit.

• 2f_{1}(x) = 2

*x−*1 =*f*_{2}(x)

Graph of *f*2 is stretched vertically.

• *−f*2(x) = *−* 2

*x−*1 =*f*3(x)

Graph of *f*2 is reflected through the *x-*
axis.

• *f*3(x) + 1 = 1*−* 2

*x−*1 =*f(x)*
Graph of *f*3 moves up 1 unit.

### 1 1

1 pt

2 pt

3 pt

4 pt

5 pt

## Limit

### Practice problems

### Limits from Graph

Exercise 1. Investigating the graph, determine the following limits.

(a) lim

*x→−1*^{+}*f*(x)
(b) lim

*x**→*0^{−}*f(x)*
(c) lim

*x**→*0*f*(x)

(d) lim

*x**→*1*f(x)*
(e) lim

*x→2*^{+}*f(x)*
(f) lim

*x→2**f(x)*

-1 1 2

1

*y* =*f*(x)

*x*
*y*

Solution.

(a) As *x* approaches *−*1 through the values *−*0.9, *−*0.99, *−*0.999..., the function approaches
the value 1. Therefore lim

*x**→−*1^{+}*f(x) = 1.*

(b) As *x* approaches 0 through the values *−*0.1, *−*0.01, *−*0.001..., the function approaches
the value 0. Therefore lim

*x→0*^{−}*f*(x) = 0.

(c) As*x*approaches 0 through either positive or negative values, the function approaches the
value 0. Therefore lim

*x**→*0*f*(x) = 0 even if *f* is defined at *x*= 0 as *f(0) = 1.*

(d) We can see that lim

*x**→*1*f*(x) does not exist because the left-handed limit, which is clearly 1,

*x*lim*→*1^{−}*f(x) = 1, is different from the right-handed limit, which is 0, lim*

*x**→*1^{+}*f(x) = 0.*

(e) Right-handed limit lim

*x**→*2^{+}*f(x) is not defined because the domain of* *f* is the [*−*1,2] closed
interval, that is the function is not defined for values grater than 2.

(f) Since 2 is the right hand end point of the domain of*f*, we have lim

*x**→*2*f*(x) = lim

*x→2*^{−}*f*(x) = 0.

27

### Limits by Substitution

Exercise 2. Find the following limits.

(a) lim

*x→5*(x*−*3)
(b) lim

*x**→−*3(x^{2}+ 4x+ 3)
(c) lim

*y**→−*2

*y*^{2}+ 5

(d) lim

*x→−1*(π*−*1)
(e) lim

*v→2*

*v*^{3}+ 8
*v*^{4} *−*6

(f) lim

*v**→−*2

*v*^{3}+ 8
*v*^{4}*−*6
(g) lim

*z**→*3(2^{z}*−*1)
(h) lim

*t**→*2log_{2}(4t*−*7)
Solution. To calculate limit of a function, we always start with substitution.

(a) Substituting 5 for *x* we have

*x→5*lim(x*−*3) = 5*−*3 = 2.

(b) Substituting *−*3 for *x* we have

*x*lim*→−*3(x^{2}+ 4x+ 3) = (*−*3)^{2}+ 4*·*(*−*3) + 3 = 9*−*12 + 3 = 0.

(c) Similarly,

*y*lim*→−*2

*y*^{2}+ 5 =

(*−*2)^{2}+ 5 =*√*
9 = 3.

(d) Since there is no *x* in the expression *π−*1, for any *x* value the function value is *π−*1,
that is we have a constant function, so

*x*lim*→−*1(π*−*1) = *π−*1.

(e)

lim*v**→*2

*v*^{3} + 8

*v*^{4}*−*6 = 8 + 8
16*−*6 = 16

10 (f)

*v*lim*→−*2

*v*^{3}+ 8

*v*^{4}*−*6 = *−*8 + 8
16*−*6 = 0
(g)

*z*lim*→*3(2^{z}*−*1) = 2^{3}*−*1 = 7
(h)

lim*t**→*2log_{2}(4t*−*7) = log_{2}1 = 0

### Eliminating Zero Denominators Algebraically

Exercise 3. Find the following limits.

(a) lim

*x→2*

(x*−*3)(x*−*2)
*x−*2
(b) lim

*x→−2*

*x*^{2}+ 4x+ 4
*x*+ 2
(c) lim

*y**→*3

*y*^{2}*−*9
*y*^{2}*−*3y

(d) lim

*z**→*1

*z*^{2}*−*2z+ 1
*z*^{2} +*z−*2
(e) lim

*t**→*1

*√t−*1
*t−*1

(f) lim

*h**→*0

*h*
1*−√*

*h*+ 1
(g) lim

*v→4*

*√v−*2
*v*^{2}*−*16

Solution. If substitution does not give a valid number, then we have to figure out something else. When after substitution both the numerator and the denominator have a value 0, pro- ducing the meaningless fractional form 0/0, we try to cancel common factors or rationalize the numerator and/or the denominator, and reevaluate the simplified form by direct substitution.

(a) Substituting 2 for*x*we have the form 0/0. However, in determining the limit at 2 we are
concerned with only the values of the function when *x* is close to 2, but*x* = 2. Thus we
can divide numerator and denominator by *x−*2 to obtain

*x*lim*→*2

(x*−*3)(x*−*2)

*x−*2 = lim

*x**→*2(x*−*3).

We can calculate this new limit by direct substitution to have

*x*lim*→*2

(x*−*3)(x*−*2)
*x−*2 = lim

*x**→*2(x*−*3) = 2*−*3 = *−*1.

(b) By direct substitution we obtain the form 0/0. Therefore both the numerator and the
denominator have a common factor of (x+ 2). Since *x*^{2}+ 4x+ 4 = (x+ 2)^{2}, we have

*x→−2*lim

*x*^{2}+ 4x+ 4

*x*+ 2 = lim

*x→−2*

(x+ 2)^{2}

*x*+ 2 = lim

*x→−2*(x+ 2) =*−*2 + 2 = 0.

(c) Substitution yields form 0/0. Because of *y*^{2}*−*9 = (y*−*3)(y+ 3), *y*^{2}*−*3y=*y(y−*3), we
can cancel common factors:

*y*lim*→*3

*y*^{2} *−*9

*y*^{2}*−*3y = lim

*y**→*3

(y*−*3)(y+ 3)

*y(y−*3) = lim

*y**→*3

*y*+ 3

*y* = 3 + 3
3 = 2.

(d) Substitution gives 0/0.

*z*lim*→*1

*z*^{2}*−*2z+ 1

*z*^{2}+*z−*2 = lim

*z**→*1

(z*−*1)^{2}

(z+ 2)(z*−*1) = lim

*z**→*1

*z−*1
*z*+ 2 = 0

(e) Form 0/0. In this case, first we rewrite the fraction by rationalizing the numerator.

lim*t**→*1

*√t−*1
*t−*1 = lim

*t**→*1

*√t−*1
*t−*1 *·*

*√t*+ 1

*√t*+ 1 = lim

*t**→*1

(*√*

*t−*1)*·*(*√*
*t*+ 1)
(t*−*1)*·*(*√*

*t*+ 1)

= lim

*t→1*

*t−*1
(t*−*1)*·*(*√*

*t*+ 1) = lim

*t→1*

*√* 1

*t*+ 1 = 1
2

(f) Form 0/0. We rewrite the fraction by rationalizing the denominator.

*h*lim*→*0

*h*
1*−√*

*h*+ 1 = lim

*h**→*0

*h*
1*−√*

*h*+ 1 *·* 1 +*√*
*h*+ 1
1 +*√*

*h*+ 1 = lim

*h**→*0

*h(1 +√*
*h*+ 1)
1*−*(h+ 1)

= lim

*h**→*0

*h(1 +√*
*h*+ 1)

*−h* = lim

*h**→*0

1 +*√*
*h*+ 1

*−*1 = 2

*−*1 =*−*2
(g) Form 0/0.

*v→4*lim

*√v−*2

*v*^{2}*−*16 = lim

*v→4*

*√v−*2
*v*^{2}*−*16 *·*

*√v*+ 2

*√v*+ 2 = lim

*v→4*

*v−*4
(v^{2}*−*16)*·*(*√*

*v*+ 2)

= lim

*v**→*4

*v−*4
(v*−*4)(v + 4)(*√*

*v*+ 2) = lim

*v**→*4

1
(v+ 4)(*√*

*v*+ 2) = 1
32

### One-sided Limits

Exercise 4. Find the following limits.

(a) lim

*x**→*0

2x*−*1
*x*
(b) lim

*x→2*

1*−x*
*x(2−x)*

(c) lim

*x**→*0

3
*x*^{2}(x*−*2)
(d) lim

*x→−1*

*x*^{2}+ 1
*x*^{2}*−*3x*−*4

Solution. When after substitution only the denominator has a value 0 the limit is either *∞*,

*−∞* or does not exist. We calculate one-sided limits to decide on that.

(a) By direct substitution we obtain the form *−*1/0. Hence, we are looking at the one-sided
limits. First consider

*x*lim*→*0^{+}

2x*−*1
*x* *.*

As *x* approaches 0 from the right, the numerator is eventually a negative quantity ap-
proaching*−*1 and the denominator a positive quantity approaching 0. Consequently, the
ratio is a negative quantity that decreases without bound. This means

*x→0*lim^{+}

2x*−*1

*x* =*−∞.*
Now consider

*x*lim*→*0^{−}

2x*−*1
*x* *.*

As*x*approaches 0 from the left, the numerator is a negative quantity approaching*−*1 and
the denominator a negative quantity approaching 0. Consequently, the ratio is a positive
quantity that increases without bound. That is,

*x*lim*→*0^{+}

2x*−*1
*x* =*∞.*
The side-limits are not the same, so

*x→0*lim

2x*−*1
*x*
does not exist.

(b) Substitution yields the form *−*1/0. Let us write the function as
1*−x*

*x(2−x)* = 1*−x*

*x* *·* 1

2*−x*

to separate the factor responsible for the 0 in the denominator. Then, by

*x*lim*→*2

1*−x*
*x* = *−*1

2 *,* lim

*x→2*^{+}

1

2*−x* =*−∞,* lim

*x→2*^{−}

1

2*−x* =*∞*
we have

*x*lim*→*2^{+}

1*−x*

*x(2−x)* =*∞,* lim

*x**→*2^{−}

1*−x*

*x(2−x)* =*−∞.*
Therefore

*x→2*lim

1*−x*
*x(2−x)*
does not exist.

(c) Substitution gives 3/0. By

3

*x*^{2}(x*−*2) = 3
*x−*2 *·* 1

*x*^{2}
and

*x*lim*→*0

3

*x−*2 = 3

*−*2*,* lim

*x→0*^{+}

1

*x*^{2} =*∞,* lim

*x→0*^{−}

1
*x*^{2} =*∞*
we have

*x*lim*→*0^{+}

1*−x*

*x(2−x)* =*−∞,* lim

*x**→*0^{−}

1*−x*

*x(2−x)* =*−∞.*
It means, that

*x*lim*→*0

3

*x*^{2}(x*−*2) =*−∞.*
(d) Form 2/0. By

*x*^{2}+ 1

*x*^{2}*−*3x*−*4 = *x*^{2}+ 1
*x−*4 *·* 1

*x*+ 1
and

*x*lim*→−*1

*x*^{2}+ 1
*x−*4 = 2

*−*5*,* lim

*x**→−*1^{+}

1

*x*+ 1 =*∞,* lim

*x**→−*1^{−}

1

*x*+ 1 =*−∞,*
we have

*x**→−*lim1^{+}

*x*^{2}+ 1

*x*^{2}*−*3x*−*4 =*−∞,* lim

*x**→−*1^{−}

*x*^{2}+ 1

*x*^{2}*−*3x*−*4 =*∞.*
Therefore

*x*lim*→−*1

*x*^{2}+ 1
*x*^{2}*−*3x*−*4
does not exist.

### Supplementery exercises

### Limits from Graph

Exercise 1. Using the graph, determine the following limits.

(a) lim

*x**→−*1^{+}*f*(x)
(b) lim

*x→0*^{−}*f(x)*
(c) lim

*x→0**f*(x)

(d) lim

*x→1**f(x)*
(e) lim

*x**→*2^{+}*f(x)*
(f) lim

*x**→*2*f(x)*

-1 1 2

1

*y*= *f*(x)

*x*
*y*

Solution.

(a) 0 (b) 1

(c) Does not exist.

(d) 1

(e) Not defined.

(f) 0

### Limits by Substitution

Exercise 2. Find the following limits and simplify your result.

(a) lim

*x→−7*(2x+ 5)
(b) lim

*x→2*(*−x*^{2}+ 5x*−*2)
(c) lim

*t→6*(t*−*5)(t*−*7)

(d) lim

*x**→*2

*x*+ 3
*x*+ 6
(e) lim

*x**→*5

4
*x−*7
(f) lim

*y→−5*

*y*^{2}
5*−y*

(g) lim

*z→0*(2z*−*8)^{1/3}
(h) lim

*t→−3*(5*−t)*^{4/3}
(i) lim

*h**→*0

*√* 3

3h+ 1 + 1 Solution.

(a) *−*9
(b) 4

(c) *−*1

(d) 5/8
(e) *−*2
(f) 5/2

(g) *−*2
(h) 16

(i) 3/2

### Eliminating Zero Denominators

Exercise 3. Find the following limits.

(a) lim

*x**→*5

*x−*5
*x*^{2}*−*25
(b) lim

*x→−5*

*x*^{2}+ 3x*−*10
*x*+ 5
(c) lim

*x**→−*3

*x*+ 3
*x*^{2}+ 4x+ 3
(d) lim

*t**→*1

*t*^{2} +*t−*2
*t*^{2} *−*1

(e) lim

*t**→−*1

*t*^{2}+ 3t+ 2
*t*^{2}*−t−*2
(f) lim

*y**→*0

5y^{3}+ 8y^{2}
3y^{4}*−*16y^{2}
(g) lim

*u**→*1

*u*^{4}*−*1
*u*^{3}*−*1
(h) lim

*x→9*

*√x−*3
*x−*9

(i) lim

*h**→*0

*√*3h+ 1*−*1
*h*
(j) lim

*h→0*

*√*5h+ 4*−*2
*h*
(k) lim

*x→4*

4x*−x*^{2}
2*−√*

*x*
(l) lim

*y**→*1

*y−*1

*√y*+ 3*−*2
Solution.

(a) _{10}^{1}
(b) *−*7

(c) *−*^{1}_{2}
(d) ^{3}_{2}

(e) *−*^{1}_{3}
(f) *−*^{1}_{2}
(g) ^{4}_{3}
(h) ^{1}_{6}

(i) ^{3}_{2}
(j) ^{5}_{4}
(k) 16

(l) 4

### One-sided Limits

Exercise 4. Find the following limits.

(a) lim

*x→2*^{−}

3
*x−*2
(b) lim

*x**→*2^{+}

3
*x−*2
(c) lim

*x→2*

3
*x−*2

(d) lim

*t**→*3

1
*t−*3
(e) lim

*z**→−*8^{+}

2z
*z*+ 8
(f) lim

*z**→−*8^{−}

2z
*z*+ 8

(g) lim

*y**→*7

4
(y*−*7)^{2}
(h) lim

*p**→−*2^{+}

2*−p*
*p*^{2}+*p−*2
(i) lim

*q**→*1^{−}

2 +*q*
*q*^{2}*−*3q+ 2
Solution.

(a) *−∞*

(b) *∞*

(c) Does not exist.

(d) Does not exist.

(e) *−∞*

(f) *∞*

(g) *∞*
(h) *−∞*

(i) *∞*

### Mixed exercises

Exercise 5. Find the following limits.

(a) lim

*u**→*2

*u−*2
*u*^{2}*−*4
(b) lim

*u**→−*2

*u−*2
*u*^{2}*−*4

(c) lim

*v**→*3

*v*+ 3
*v*^{2} *−*6v+ 9
(d) lim

*v**→*3

*v−*3
*v*^{2} *−*6v+ 9

(e) lim

*h→0*

(x+*h)*^{2}*−x*^{2}
*h*
(f) lim

*x**→*0

(x+*h)*^{2}*−x*^{2}
*h*
Solution.

(a) ^{1}_{4}

(b) Does not exist.

(c) *∞*

(d) Does not exist.

(e) 2x
(f) *h*

### Quizzes

*Version A*

Exercise. Find the following limits.

(a) lim

*x**→−*2

*x*+ 2
*x*^{2}+ 4
(b) lim

*x**→*5

*x*^{2}*−*3x*−*10
*x−*5
(c) lim

*y→4*

*√y−*2
*y*^{2}*−*4y
(d) lim

*t**→*3

*t−*2
*t*^{2} *−t−*6
*Version B*

Exercise. Find the following limits.

(a) lim

*p**→*3

*p−*3
*p*^{2}+ 3p
(b) lim

*x**→−*2

*x*^{2}+ 2x
*x*^{2}+*x−*2
(c) lim

*x→2*

*√x*+ 2*−*2
*x*^{2}*−*3x+ 2
(d) lim

*t**→*3

2t*−*2
*t*^{2} *−*5t+ 6