CALCULUS I
Chapter 1: Functions, Limits and Continuity
Assoc. Prof. Nguyen Ngoc Hai
DEPARTMENT OF MATHEMATICS
September 27, 2022
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
References
Main textbook:
J. Steward, Calculus. Early Transcendentals, 8th ed.,
Thomson Learning, 2016.
Other textbook:
J. Rogawski, C. Adams, R. Franzosa Calculus, Early
Transcendentals, W. H. Freeman, 2018.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Chapter 1. FUNCTIONS, LIMITS
AND CONTINUITY
Contents
1. What is Calculus?
2. Straight Lines. Equations of Lines
3. Functions and Graphs
4. New Functions from Old Functions. Inverse
Functions
5. Parametric Curves
6. Limits of Functions. One-sided Limits
7. Laws of Limits. Evaluating Limits
8. Continuity. The Intermediate Value Theorem
9. Limits Involving Infinity
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Chapter 1 FUNCTIONS, LIMITS
AND CONTINUITY
1.1
WHAT IS CALCULUS?
Early in the seventeenth century, Johannes Kepler
(1571-1630) discovered three laws of planetary
motion:
1. Each planet travels in an ellipse that has one
focus at the sun.
2. The radius vector from the sun to a planet
sweeps out equal areas in equal intervals of time.
3. If T is the length of a planet’s year and a is the
semimajor axis of its orbit, then the ratio T 2 /a3
has the same constant value for all planets in
the solar system.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Chapter 1 FUNCTIONS, LIMITS
AND CONTINUITY
1.1
WHAT IS CALCULUS?
Questions:
1. Why do the planets move in elliptical orbits
around the sun?
2. How do radio waves propagate through space?
3. How can one predict the effects of interest rate
changes on economies and stock markets?
4. Why does an epidemic spread faster and faster
and then slow down?
♠ These and many other questions of interest and
importance in our world relate directly to our ability
to analyze motion and how quantities change with
respect to time or each other.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Chapter 1 FUNCTIONS, LIMITS
AND CONTINUITY
1.1
WHAT IS CALCULUS?
• Kepler described how the solar system worked.
He didn’t know why.
• Calculus and Newton’s laws explained why it
worked that way.
• Algebra and geometry are useful tools for
describing relationships among static quantities,
but they do not involve concepts appropriate for
describing how a quantity changes.
• Calculus provides the tools for describing motion
quantitatively. It introduces two new operations
called differentiation and integration.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Chapter 1 FUNCTIONS, LIMITS
AND CONTINUITY
1.1
WHAT IS CALCULUS?
• Differential calculus dealt with the problem of
calculating rates of change.
• Integral calculus dealt with the problem of
determining a function from information about
its rate of change.
• Calculus is the mathematics of motion and
change.
• John von Neumann (1903-1957) wrote: “The
calculus was the first achievement of modern
mathematics”.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
HOW TO LEARN CALCULUS?
Calculus introduces so many new concepts and
computational operations.
What should you do to learn?
1. Read the text carefully. Read the relevant
passages in the textbook and work through the
examples step by step. Read and search for
detail in a step by step logical fashion. It takes
attention, patience, and practice.
2. Complete the homework exercises, keeping the
following principles in mind.
(a) Sketch a diagram whenever possible.
(b) Write your solution in a connected step-by-step logical fashion,
as if you were explaining to someone else.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Chapter 0 PRELIMINARIES
3. Finally, try on your own to write short
descriptions of the key points each time you
complete a section of the text.
(G. B. Thomas, Jr., R. L. Finney, Calculus and
Analytic Geometry, 9th ed., Addison-Wesley, 1998)
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.1
STRAIGHT LINES
Linear functions are the simplest of all functions and
their graphs (lines) are the simplest of all curves.
However, linear functions and lines play an
enormously important role in calculus.
For this reason, it is important to be thoroughly
familiar with the basic properties of linear functions
and the different ways of writing the equations of a
line.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.1
STRAIGHT LINES
Slopes of Nonvertical Lines
If a particle moves from (x1 , y1 ) to (x2 , y2 ), the
increments in its coordinates are
∆x = x2 − x1
and
∆y = y2 − y1 .
Let L be a nonvertical line in the plane.
Let P1 (x1 , y1 ) and P2 (x2 , y2 ) be points on L.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.1
STRAIGHT LINES
Definition 2.1
The slope of a nonvertical line is
m=
∆y
y2 − y1
=
∆x
x2 − x1
• The slope of a horizontal line is zero since
∆y = 0.
• The slope of a vertical line is undefined.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.2
EQUATIONS OF LINES
• To write an equation for a line that is not
vertical, it is enough to know its slope m and
the coordinates of a point P1 (x1 , y1 ) on it.
Definition 2.2
The equation
y − y1 = m(x − x1 )
is the point-slope equation of the line that passes
through the point (x1 , y1 ) with slope m.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.2
EQUATIONS OF LINES
Example 2.1 Write an equation for the line that
passes through the point (2, 3) with slope −3/2.
Example 2.2 Write an equation for the line
through (−2, −1) and (3, 4).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.2
EQUATIONS OF LINES
Slope-Intercept Equations
Definition 2.3
The equation
y = mx + b
is the slope-intercept equation of the line with
slope m and y -intercept b.
(See Linear Function Explorer,
www.mathopenref.com/linearexplorer.html)
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.2
EQUATIONS OF LINES
Example 2.3 The standard equation for
converting Celsius temperature to Fahrenheit
temperature is a slope-intercept equation. If 0◦ C
corresponds to 32◦ F (the freezing point of water)
and 100◦ C corresponds to 212◦ F (the boiling point
of water at see level), represent Fahrenheit
temperature F as a function of Celsius temperature
C.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.2
EQUATIONS OF LINES
Parallel and Perpendicular Lines
Parallel lines have equal angles of inclination. Thus,
♠Two lines are parallel if and only if they have
the same slope, or if they are both vertical.
♠Two lines are perpendicular if and only if the
product of their slopes is −1 or, if one is vertical
and the other horizontal.
FIGURE 1.1
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.2 STRAIGHT LINES. EQUATIONS OF LINES
1.2.2
EQUATIONS OF LINES
Example 2.4 Find the equation of the line that
passes through the point (3, 5) and is parallel to the
line 2x + 5y = 4.
Example 2.5 Find the slope of any line L
perpendicular to the line having the equation
5x − y = 4.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
In many practical situations, the value of one
quantity may depend on the value of a second.
• For example, the area A of a circle depends on
the radius r of the circle. The rule that connects
A and r is given by A = πr 2 .
• The human population of the world P depends
on the time t. For instance,
P(1980) = 4.45 billions,
P(2000) = 6.070 billions.
Such relationships can often be represented
mathematically as functions.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
Definition 3.1
A function from a set A to a set B is a rule that
assigns to each element in A a single element of B.
The set A is called the domain of the function.
The set of all possible values of the function is
called the range.
FIGURE 1.2 A function f : D → E
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
• We usually consider functions for which the sets
A and B are sets of real numbers.
• To denote that y is a function of x we write
y = f (x).
The number f (x) is the value of
Assoc. Prof. Nguyen Ngoc Hai
f at x .
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
• A symbol that represents an arbitrary number in
the domain of a function f is called an
independent variable.
• A symbol that represents a number in the range
of f is called a dependent variable.
So,
The set of all possible values of the independent
variable in a function is its domain, and the
resulting set of all possible values of the dependent
variable is the range.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
Example 3.1
functions?
(a)
Which of the following are
(b) The key x 2 on a calculator.
(c) The set of order pairs with first elements
children and second elements their birth mothers.
(d) The set of order pairs with first elements
mothers and second elements their children.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
Representations of functions
There are four possible ways to represent a function
•
•
•
•
verbally
numerically
visually
algebraically
(by
(by
(by
(by
a description in words)
a table of values)
a graph)
an explicit formula)
For instance, the most useful representation of the
area of a circle as a function of its radius is probably
the algebraic formula A = πr 2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
• In most cases in this course, a function is
expressed as an equation, such√as
C (x) = 5x − 2 + x 2 − 1.
• When an equation is given for a function, we say
that the equation defines the function.
Example 3.2
Let g (x) = −x 2 + 4x − 5.
Find each of the following
(a) g (3),
(b) g (a),
(c) g (x + h),
(d) g ( 2r ),
(e) Find all values of x such that g (x) = −2.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
• A function is not properly defined until its
domain is specified.
Agreement on Domains
When a function f is defined without specifying its
domain, we assume that the domain consists of all
real numbers x for which the value f (x) of the
function is a real number.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
Example 3.3
Find the domain and range for
each of the functions defined as follows.
√
(a) f (x) = x 2 − 5x + 6 ,
(b) g (t) =
t
.
t2 − 1
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
Piecewise Defined Functions
Sometimes it is necessary to define a function by
using different formulae on different parts of its
domain.
Example 3.4 The absolute value function
f (x) = |x| is defined by
(
x if x ≥ 0
f (x) = |x| =
−x if x < 0.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.1
FUNCTIONS
Example 3.5
The signum function is defined by


 1 if x > 0
sgn (x) =
0 if x = 0


−1 if x < 0.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
Graphs of Equations
Each point in the plane corresponds to an ordered
pair of numbers.
• The first member is called the first coordinate
of the point, and the second member is called
the second coordinate. Together, these are
called the coordinates of the point.
• In the xy -plane, the vertical line is often called
the y -axis, and the horizontal line is often
called the x -axis.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
Definition 3.2
The graph of an equation is a drawing that
represents all the solutions of the equation.
For instance, the graph of the equation
(x − x0 )2 + (y − y0 )2 = R 2
(R > 0)
is the circle with center at (x0 , y0 ) and radius R.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
Graphs of Functions
Definition 3.3
If f is a function with domain A, then its graph is
the set of all order pairs
x, f (x) | x ∈ A
In other words, the graph of f consists of all points
(x, y ) in the xy -plane such that y = f (x) and x is
in the domain of f .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
FIGURE 1.3 The graph of f (x) = |x|
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
Example 3.6
Sketch the graph of
(a) x 2 + y 2 = 4,
(b) y = x 2 ,
(c) x = y 2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
The graph of a function is a curve in the xy -plane.
♠ Question:
Which curves in the xy -plane are graphs of
functions?
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.2
GRAPHS OF EQUATIONS AND FUNCTIONS
The Vertical Line Test
A curve in the xy -plane is the graph of a function of
x if and only if no vertical line intersects the curve
more than one.
FIGURE 1.4
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.3
EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
Definition 3.4
Suppose that −x belongs to the domain of f
whenever x does.
• We say that f is an even function if
f (−x) = f (x) for every x in the domain of f .
• We say that f is an odd function if
f (−x) = −f (x) for every x in the domain of f .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.3
EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
• The graph of an even function is symmetric
about the y axis.
FIGURE 1.5
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.3
EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
• The graph of an odd function is symmetric
about the origin. If an odd function f is defined
at x = 0, then f (0) = 0.
FIGURE 1.6
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.3
EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
Example 3.7 Determine whether the function is
even, odd, or neither.
(a) f (x) = x 6 ,
(b) g (x) = x1 ,
(c) h(x) = x 3 + x 2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.3
EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
Increasing and Decreasing Functions
Definition 3.5
A function f is called increasing on an interval I if
f (x1 ) < f (x2 ) for all x1 , x2 ∈ I such that x1 < x2 .
It is called decreasing on I if
f (x1 ) > f (x2 ) for all x1 , x2 ∈ I such that x1 < x2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.3 FUNCTIONS AND GRAPHS
1.3.3
EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
FIGURE 1.7
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
Two important ways of modifying a graph are
translation (or shifting) and scaling.
Translation consists of moving the graph
horizontally or vertically.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
Vertical and horizontal translation
Suppose c > 0. To obtain the graph of
• y = f (x) + c, shift the graph of y = f (x) a
distance c units upward;
• y = f (x) − c, shift the graph of y = f (x) a
distance c units downward;
• y = f (x − c), shift the graph of y = f (x) a
distance c units to the right;
• y = f (x + c), shift the graph of y = f (x) a
distance c units to the left.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
FIGURE 1.8
Note that f (x) + c and f (x + c) are different.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
Example 4.1
Given the graph of
y = f (x) =
1
,
x2 + 1
use transformation to graph
x2 + 2
,
y= 2
x +1
−2x 2 − 1
y=
,
x2 + 1
y=
1
.
(x + 1)2 + 1
Example 4.2 Sketch the graph of the function
f (x) = x 2 + 4x − 5 using the graph of y = x 2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
Given a line L and a point P not on L, we call a
point Q the reflection of P in L if L is the right
bisector of the line segment PQ.
The reflection of any graph G in L is the graph
consisting of the reflections of all of the points of G.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
Vertical and Horizontal Stretching and Reflecting
Suppose that c > 1. To obtain the graph of
y = cf (x), stretch the graph of y = f (x) vertically by a
factor of c;
y = c1 f (x), compress the graph of y = f (x) vertically by a
factor of c;
y = f (cx), compress the graph of y = f (x) horizontally
by a factor of c;
y = f ( xc ), stretch the graph of y = f (x) horizontally by a
factor of c;
y = −f (x), reflect the graph of y = f (x) about the
x-axis;
y = f (−x), reflect the graph of y = f (x) about the
y -axis.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
FIGURE 1.9
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.1
TRANSFORMATIONS OF FUNCTIONS
Example 4.3
functions
Sketch the graph of the following
(a) y = sin 5x,
(b) y = 3 − sin 2x,
(c) y = | ln x|.
♠ Note
Remember that cf (x) and f (cx) are different.
The graph of y = cf (x) is a vertical scaling and
y = f (cx) a horizontal scaling of the graph of
y = f (x).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.2
SUMS, DIFFERENCES, PRODUCTS QUOTIENTS, AND MULTIPLES
Definition 2.1
If f and g are functions, then for every x that belongs to the
domains of both f and g we define functions f + g , f − g , fg ,
f /g by the formulas:
(f + g )(x) = f (x) + g (x)
(f − g )(x) = f (x) − g (x)
(fg )(x) = f (x)g (x)
f
f (x)
(x) =
,
where g (x) ̸= 0.
g
g (x)
In particular, if c is a real number, then the function cf is
defined for all x in the domain of f by (cf )(x) = c · f (x).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.2
SUMS, DIFFERENCES, PRODUCTS QUOTIENTS, AND MULTIPLES
Example 4.4 If f (x) =
find the functions
6f ,
f + g,
√
f − g,
x and g (x) =
fg ,
and
√
4 − x 2,
f
,
g
and specify the domains of each of these functions.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.3
COMPOSITE FUNCTIONS
Definition 4.2
Given two functions f and g , the composite
function f ◦ g (also called the composition of
and g ) is given by
f ◦ g (x) = f g (x) .
f
FIGURE 1.10
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.3
COMPOSITE FUNCTIONS
Note
The notation f ◦ g means that the function g is
applied first and then f is applied second.
The domain of f ◦ g is the set of all x in the
domain of g for which g (x) is in the domain of f .
If the range of g is contained in the domain of f
then the domain of f ◦ g is just the domain of g .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.3
COMPOSITE FUNCTIONS
Example
√ 4.5 If f (x) = x + 1 and
g (x) = 4 − x 2 , calculate the four composite
functions f ◦ g , f ◦ f , g ◦ g , and g ◦ f , and specify
the domain of each.
♠ Note
In general f ◦ g ̸= g ◦ f .
It is possible to take the composition of three or
more functions.
p
Example 4.6 Given F (x) = 2 + cos(x 2 + 1) ,
find functions f , g and h such that F = f ◦ g ◦ h.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.4
THE BASIC CLASSES OF FUNCTIONS
Polynomials
Definition 4.3
For any real number α, the function f (x) = x α is called the
power function with exponent α.
A function P is called a polynomial if
P(x) = an x n + an−1 x n−1 + · · · + a1 x + a0
where n is a nonnegative integer number and the numbers
an , an−1 , . . . , a0 are constants.
The numbers an , an−1 , ..., a0 are called coefficients.
The degree of P is n (assuming that an ̸= 0).
The coefficient an is called the leading coefficient.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.4
THE BASIC CLASSES OF FUNCTIONS
• A polynomial of degree 1 is of the form
f (x) = ax + b and so it is a linear function.
• A polynomial of degree 2 is called a quadratic
function. Its graph is always a parabola.
• A polynomial of degree 3 is of the form
P(x) = ax 3 + bx 2 + cx + d
(a ̸= 0)
and is called a cubic function.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.4
THE BASIC CLASSES OF FUNCTIONS
Rational Functions
Definition 4.4
A rational function is a quotient of two
polynomials
P(x)
f (x) =
.
Q(x)
Every polynomial is also a rational function (with
Q(x) = 1).
The domain of a rational function
{x| Q(x) ̸= 0}.
Assoc. Prof. Nguyen Ngoc Hai
P(x)
Q(x)
is the set
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.4
THE BASIC CLASSES OF FUNCTIONS
Algebraic Functions
Definition 4.5
A function is called an algebraic function if it can
be constructed using algebraic operations (such as
addition, substraction, multiplications, division, and
taking roots) starting with polynomials.
For example,
√
x7 − x2 + 3
√
f (x) =
+ (x − 1) 6 2x + 1
x4 − x2 − 1
is an algebraic functions.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
Definition 4.7
A function f is called a one-to-one function if
f (x1 ) ̸= f (x2 ) whenever x1 and x2 belong to the
domain of f and x1 ̸= x2 .
In other words, a function is one-to-one if it never
takes on the same values twice, that is, fore every
value c, the equation f (x) = c has at most one
solution for x.
An equivalent statement is that
f is one-to-one if f (x1 ) = f (x2 ) =⇒ x1 = x2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
The Horizontal Line Test
A function is one-to-one if and only if no horizontal
line intersects its graph more than once.
FIGURE 1.11
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
Example 4.7
one-to-one?
Which of the following functions is
(a) f (x) = x 2 ;
(b) g : [0, ∞) → R,
g (x) = x 2
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
Definition 4.8
Let f be a one-to-one function with domain D and
range E . Then its inverse function f −1 has
domain E and range D and defined by
f −1 (y ) = x ⇐⇒ f (x) = y
for any y in E .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
FIGURE 1.12
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
Note that
domain of f −1 = range of f
range of f −1 = domain of f
Example 4.8
Find the inverse of
√
f (x) = 2x + 1 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
How to find the inverse function of f ?
1. Solve the equation y = f (x) for x in terms of y
(if possible).
2. Interchange x and y . The resulting equation will
be y = f −1 (x).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.4 NEW FUNCTIONS FROM OLD FUNCTIONS
INVERSE FUNCTIONS
1.4.5
INVERSE FUNCTIONS
f −1 f (x) = x
f f −1 (x) = x
for every x in D
for every x in E .
The graph of f −1 is obtained by reflecting the
graph of f about the line y = x.
FIGURE 1.13
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
Exponential and Logarithmic Functions
An exponential function is a function of the form
f (x ) = ax , where a > 0 and a ̸= 1. The number a
is called the base.
• Exponential functions are positive: ax > 0 for all
x.
• f (x) = ax is increasing if a > 1 and decreasing
if a < 1.
• The domain of an exponential function is
R = (−∞, ∞) and the range is (0, ∞).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
Definition 4.9
If a > 0 and a ̸= 1, then the logarithm to the
base a, denoted loga x , is the inverse of f (x) = ax .
y = loga x ⇐⇒ x = ay
• The domain of loga x is (0, ∞).
• The range of loga x is the set of all real number
R.
• f (x) = loga x is increasing if a > 1 and
decreasing if a < 1.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
FIGURE 1.14 The graphs of y = e x and y = loga x
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
Inverse Trigonometric Functions
• The function f (x) = sin x is one-to-one on
[−π/2, π/2]. --> half the circle
• Its inverse is called the inverse sine function
or the arcsine function and denoted sin−1 x or
arcsin x .
h π πi
−1
such
y = sin x is the unique angle in − ,
2 2
that sin y = x.
y = sin−1 x ⇐⇒ sin y = x
Assoc. Prof. Nguyen Ngoc Hai
and
−1≤x ≤1
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
• The domain of sin−1 x is [−1, 1].
• The range is [−π/2, π/2].
FIGURE 1.15 The graph of y = sin−1 x = arcsin x
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
• The cosine function is one-to-one on [0, π].
• Its inverse is called the inverse cosine
function or the arccos function and denoted
cos−1 x or arccos x .
y = cos−1 x is the unique angle in [0, π] such that
cos y = x.
y = cos−1 x ⇐⇒ cos y = x
Assoc. Prof. Nguyen Ngoc Hai
and
−1≤x ≤1
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
• The domain of cos−1 x is [−1, 1].
• The range is [0, π].
FIGURE 1.16 The graph of y = cos−1 x = arccos x
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
• The tangent function is one-to-one on the
interval (−π/2, π/2).
• The inverse is called the inverse tangent
function and is denoted tan−1 x or arctan x :
y = tan−1 x is the unique angle in − π2 , π2 such
that tan y = x.
y = tan−1 x ⇐⇒ tan y = x
Assoc. Prof. Nguyen Ngoc Hai
and
−∞<x <∞
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
FIGURE 1.17 The graph of y = tan−1 x = arctan x
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
SOME IMPORTANT INVERSE FUNCTIONS
Similarly,
y = cot−1 x is the unique angle in (0, π) such that
cot y = x.
y = cot−1 x ⇐⇒ cot y = x
and
−∞<x <∞
• tan−1 x and cot−1 x have domain R.
• The range of tan−1 x is (−π/2, π/2).
• The range of cot−1 x is (0, π).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
ELEMENTARY FUNCTIONS
• We have reviewed some of the most basic and
familiar functions of mathematics.
• New functions may be produced using the
operations of addition, multiplication, division,
as well as composition, extraction of roots, and
taking inverses.
• It is convenient to refer to a function
constructed in this way from the basic functions
listed above as an elementary function.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
So far we have described plane curves by giving y as
a function of x (y = f (x)) or x as a function of y
(x = g (y )).
Some curves, such as the cycloid, are best handled
when both x and y are given in terms of a third
variable t called a parameter.
FIGURE 1.18 The cycloid best described by
two equations x = f (t), y = g (t)
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
Imagine that a particle moves along the curve C . It
is impossible to describe C by an equation of the
form y = f (x) or x = g (y ). But the x- and
y -coordinates of the particle are functions of time
and so we can write x = f (t) and y = g (t).
FIGURE 1.19 The orbit of an object described by two equations
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
Definition 5.1
If x and y are given as functions
x = f (t),
y = g (t)
over an interval I of t−values, then the set of points
(x, y ) = f (t), g (t) defined by these equations is a curve in
the coordinate plane. The equations are parametric
equations for the curve. The variable t is a parameter for
the curve and its domain I is the parameter interval.
If I is
a closed interval, a ≤ t ≤ b, the point f (a),
g (a) is the
initial point of the curve and f (b), g (b) is the terminal
point of the curve. When we give parametric equations and a
parameter interval for a curve in the plane, we say that we
have parametrized the curve. The equations and interval
constitute a parametrization of the curve.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
• In many applications t denotes time, but it
might instead denote an angle or the distance a
particle has traveled along its path from its
starting point.
• We could use a letter other than t for the
parameter.
• Parametric equations enable us to describe a
great variety of curves.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
FIGURE 1.20 The curve
x = 1.5 cos t − cos 30t,
y = 1.5 sin t − sin 30t.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
Example 5.1 (The Unit Circle x 2 + y 2 = 1)
What curve is represented by the parametric
equations
x = cos t,
y = sin t,
0 ≤ t ≤ 2π?
♠ Since the curve starts and ends at the same
point, it is called a closed curve.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
Example 5.2 (A parametrization of the Ellipse
y2
x2
+
2
a
b 2 = 1) Describe the motion of a particle
whose position P(x, y ) at time t is given by
x = a cos t,
y = b sin t,
0 ≤ t ≤ 2π.
Example 5.3 (A parametrization of the Circle
x 2 + y 2 = R 2 ) The equations and parameter
interval
x = R cos t,
y = R sin t,
0 ≤ t ≤ 2π
obtained by taking b = a = R in the previous
example, describe the circle x 2 + y 2 = R 2 .
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
Example 5.4 (Cycloids) A wheel of radius a
rolls (without slipping) along a horizontal straight
line. Find parametric equations for the path traced
by a point P on the wheel’s circumference. The
path is called a cycloid.
ANS. x = R(t − sin t), y = R(1 − cos t),
t ∈ R.
Note The graph of a function y = f (x) can
always be parametrized as
x =t
y = f (t).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.5
PARAMETRIC CURVES
Remark
1. There is a difference between a path and the
underlying curve C. The curve C is a set of
points in the plane. The path (the orbit)
describes not just the curve, but also the
location of a point along the curve as a function
of the parameter. The path may traverse all or
part of C several times.
2. Parametrizations are not unique, and in fact,
every curve may be parametrized in infinitely
many different ways.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Calculus was created to describe how quantities
change.
It is based on the fundamental concept of the limit
of a function.
It is this idea of limit that distinguishes calculus
from algebra, geometry, and trigonometry, which are
useful for describing static situations.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Example 6.1
function
Describe the behaviour of the
x2 − 1
f (x) =
x −1
near x = 1.
FIGURE 1.21
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Observe that we can make the value of f (x) as close
as we want to 2 by choosing x close enough to 1.
We say that f (x) approaches the limit 2 as x
approaches 1, and write
lim f (x) = 2
x→1
or
Assoc. Prof. Nguyen Ngoc Hai
x2 − 1
= 2.
lim
x→1 x − 1
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Definition 6.1
We write
lim f (x) = L
x→a
and say “the limit of f (x ) as x approaches a
equals L” if we can make the values of f (x)
arbitrarily close to L (as close to L as we like) by
taking x to be sufficiently close to a (on either side
of a) but not equal to a. We also say that f (x)
approaches L or converges to L as x approaches
a.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
An alternative notation for limx→a f (x) = L is
f (x) → L as x → a
which is usually read “f (x ) approaches
approaches a”.
Assoc. Prof. Nguyen Ngoc Hai
L as x
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Note
• In defining the limit of f (x) as x approaches a,
we never consider x = a.
• The value f (a) itself, which may or may not be
defined, play no role in the limit.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
FIGURE 1.22 limx→a f (x) = L = f (a)
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
FIGURE 1.23 limx→a f (x) = L ̸= f (a)
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
FIGURE 1.24 limx→a f (x) = L but f (a) is not defined
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Example 6.2
(a) limx→a x = a
(b) limx→a c = c (where c is a constant).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Example 6.3
Investigate
π
lim sin .
x→0
x
FIGURE 1.25 The graph of y = sin πx
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Observe that the values of sin πx oscillate between
−1 and 1 infinitely often as x approaches 0.
Since the values of f (x) do not approach a fixed
number as x approaches 0,
π
lim sin
does not exist.
x→0
x
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.2
ONE-SIDED LIMITS
The limit we have discussed so far are two-sided.
In some instances, f (x) may approach L from one
side of a without necessarily approaching it from the
other side, or f (x) may be defined on only one side
of a.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.1
LIMITS OF FUNCTIONS
Example 6.4
defined by
The Heaviside function H is
(
0 if t < 0
H(t) =
1 if t ≥ 0
As t approaches 0 from the left, H(t) approaches 0.
As t approaches 0 from the right, H(t) approaches
1. There is no single number that H(t) approaches
as t approaches 0. Therefore, limt→0 H(t) does not
exist.
FIGURE 1.26 The graph of the Heaviside function
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.2
ONE-SIDED LIMITS
Definition 6.2
We write
lim
x →a −
f (x ) = L
and say “the left-hand limit of f (x ) as x
approaches a (or the limit of f (x ) as x
approaches a from the left) equals L” if we can
make the values of f (x) as close to L as we want by
taking x to be sufficiently close to a and x less than
a. We also say that f (x) has left limit L at x = a.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.2
ONE-SIDED LIMITS
Similarly, if we require that x be greater than a, we
get “the right-hand limit of f (x ) as x
approaches a is equal to L” (or f (x) has right
limit L at x = a), and we write
lim
x →a +
f (x ) = L.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.2
ONE-SIDED LIMITS
FIGURE 1.27
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.2
ONE-SIDED LIMITS
Theorem 6.1
A function f (x) has limit L at x = a if and only if it
has both left and right limits there and these
one-sided limits are both equal to L:
lim f (x) = L ⇐⇒ lim− f (x) = lim+ f (x) = L.
x→a
x→a
Example 6.5
x→a
If
|x − 2|
,
x2 + x − 6
find limx→2+ f (x), limx→2− f (x), and limx→2 f (x).
f (x) =
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.6 LIMITS
1.6.2
ONE-SIDED LIMITS
Note
If f (x) = g (x) when x ̸= a, then
lim f (x) = lim g (x),
x→a
x→a
both
provided the limits exist.
Question What limits does g (x) =
have at x = −1 and x = 1?
Assoc. Prof. Nguyen Ngoc Hai
√
1 − x2
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.1
LAWS OF LIMITS
Theorem 7.1
Suppose that c is a constant and the limits
limx→a f (x) and limx→a g (x) exist. Then
1. limx→a f (x)+g (x) = limx→a f (x)+limx→a g (x)
2. limx→a f (x)−g (x) = limx→a f (x)−limx→a g (x)
3. limx→a cf (x) = c limx→a f (x)
4. limx→a f (x)g (x) = limx→a f (x) · limx→a g (x)
5. limx→a
limx→a f (x)
f (x)
=
if limx→a g (x) ̸= 0.
g (x) limx→a g (x)
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.1
LAWS OF LIMITS
h
in
n
6. limx→a f (x)] = limx→a f (x) ,
where n is a positive integer.
p
p
7. limx→a n f (x) = n limx→a f (x),
where n is a positive integer.
The Limit Laws also hold for one-sided limits.
Example 7.1
Find
lim
t→0
√
t2 + 9 − 3
.
t2
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.2
THE SQUEEZE THEOREM
Theorem 7.2
If f (x) ≤ g (x) when x is near a (except possibly at
a) and the limits of f and g both exist as x
approaches a, then
lim f (x) ≤ lim g (x).
x→a
Assoc. Prof. Nguyen Ngoc Hai
x→a
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.2
THE SQUEEZE THEOREM
FIGURE 1.28
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.2
THE SQUEEZE THEOREM
Theorem 7.3 (The Squeeze Theorem)
If f (x) ≤ g (x) ≤ h(x) when x is near a (except
possibly at a) and limx→a f (x) = limx→a h(x) = L,
then
lim g (x) = L.
x→a
Similar statements hold for left and right limits.
The Squeeze Theorem is sometimes called the
Sandwich Theorem or the Pinching Theorem.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.2
THE SQUEEZE THEOREM
Corollary 7.4
If limx→a |f (x)| = 0, then limx→a f (x) = 0.
Note
If |f (x)| ≤ g (x) when x is near a (except possibly
at a) and limx→a g (x) = 0, then limx→a f (x) = 0.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.2
THE SQUEEZE THEOREM
Example 7.2
Show that limx→0 x 2 sin x1 = 0.
FIGURE 1.29 The graph of y = x 2 sin x1
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.7 LAWS OF LIMITS. EVALUATING LIMITS
1.7.2
THE SQUEEZE THEOREM
Theorem 7.5
If θ is measured in radians, then
sin θ
=1
θ→0 θ
lim
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Definition 8.1
A function f is continuous at a number
a if
lim f (x) = f (a).
x→a
If f is not continuous at a, we say that f is
discontinuous at a, or f has a discontinuity at a.
Notice that if f is continuous at a, then:
1. f (a) is defined, that is, a is in the domain of f ;
2. limx→a f (x) exists;
3. limx→a f (x) = f (a).
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
FIGURE 1.30 f is continuous at a
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
The definition states that f is continuous at a if
f (x) approaches f (a) as x approaches a.
• Thus, a continuous function f has the property
that a small change in x produces only a small
change in f (x).
• In fact, the change in f (x) can be kept as small
as we please by keeping the change in x
sufficiently small.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Definition 8.2
A function f is continuous from the right at a
number a if
lim+ f (x) = f (a)
x→a
and f is continuous from the left at
a if
lim f (x) = f (a).
x→a−
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
For example, the Heaviside function
(
0 if x < 0
H(x) =
1 if x ≥ 0
is continuous at every number x except 0. It is right
continuous at 0 but is not left continuous or
continuous there.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Definition 8.3
A function f is continuous on an interval if it is
continuous at every number in the interval. If f is
continuous at all points in its domain, then f is
simply called continuous.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Here, if f is defined only on one side of an endpoint
of the interval, we understand “continuous at the
endpoint” to mean “continuous from the right” or
“continuous from the left.”
Example 8.1
continuous.
The function f (x) =
Assoc. Prof. Nguyen Ngoc Hai
√
1 − x 2 is
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Theorem 8.1
If f and g are continuous at a, and c is a constant,
then the following functions are also continuous at
a:
(a) f ± g
(b) cf
(c) fg
(d) f /g if g (a) ̸= 0.
Corollary 8.2
(a) Any polynomial is continuous everywhere, that
is, it is continuous on R = (−∞, ∞).
(b) Any rational function is continuous wherever it
is defined-that is, it is continuous on its domain.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Example 8.2 Where are each of the following
functions discontinuous?
 2
x − x − 2
if x ̸= 2
(a) f (x) =
x −2

1
if x = 2.

1
if x ̸= 0
(b) g (x) = x 2
1
if x = 0.
(
x2
if x ≤ 0
(c) h(x) =
x + 1 if x > 0.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
• The kind of discontinuity illustrated in part (a)
is called removable.
• The discontinuity in part (b) is called an
infinite discontinuity.
• The discontinuities in part (c) are called jump
discontinuities.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Theorem 8.3
If f is continuous on an interval I with range J and
if the inverse f −1 exists, then f −1 is continuous on
the domain J.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Theorem 8.4
The following types of functions are continuous at
every number in their domains:
• polynomials
• exponential functions
• rational functions • logarithmic functions
• root functions
• trigonometric functions
• inverse trigonometric functions.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Example 8.3
Where is the function
f (x) =
ln x + tan−1 x
x2 − 1
continuous?
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Theorem 8.5
If f is continuous
at b and limx→a g (x) = b, then
limx→a f g (x) = f (b). In other words,
lim f g (x) = f lim g (x) .
x→a
Assoc. Prof. Nguyen Ngoc Hai
x→a
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
CONTINUITY
Theorem 8.6
If g is continuous at a and f is continuous at g (a),
then the composite function
f ◦ g given by
(f ◦ g )(x) = f g (x) is continuous at a.
Roughly speaking,
Composites of continuous functions are continuous.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
THE INTERMEDIATE VALUE THEOREM
Theorem 8.7 (The Intermediate Value
Theorem)
A function f that is continuous on a closed interval
[a, b] takes on every value between f (a) and f (b).
FIGURE 1.31
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.8 CONTINUITY.
THE INTERMEDIATE VALUE THEOREM
1.8.1
THE INTERMEDIATE VALUE THEOREM
A point c where f (c) = 0 is called a zero or root
of f .
Corollary 8.8 (Existence of Zeros)
If f is continuous on [a, b] and if f (a) and f (b)
have opposite signs, that is, f (a)f (b) < 0, then f
has a zero in (a, b).
Example 8.4 Show that the equation
x 3 − x − 1 = 0 has a solution in the interval [1, 2].
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
Definition 9.1
Let f be a function defined on both sides of a,
except possibly at a itself. Then,
lim f (x) = ∞
x→a
means that the values of f (x) can be made
arbitrarily large-as large as we please-by taking x
sufficiently close to a, but not equal to a.
Another notation for limx→a f (x) = ∞ is:
f (x) → ∞ as x → a.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
FIGURE 1.32 limx→a f (x) = ∞
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
Definition 9.2
Let f be defined on both sides of a, except possibly
at a itself. Then,
lim f (x) = −∞
x→a
means that the values of f (x) can be made
arbitrarily large negative by taking x sufficiently
close to a, but not equal to a.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
FIGURE 1.33 limx→a f (x) = −∞
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
Similar definitions can be given for the one-sided
limits:
lim f (x) =
x→a−
∞,
lim f (x) = −∞,
x→a−
lim f (x) =
x→a+
∞,
lim f (x) = −∞.
x→a+
FIGURE 1.34
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
FIGURE 1.35 limx→0+ ln x = −∞
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
• “x → a− ” means that we consider only values of
x that are less than a.
• “x → a+ ” means that we consider only values of
x that are greater than a.
• Keep in mind that ∞ and −∞ are not numbers.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
The line x = a is called a vertical asymptote of
the curve y = f (x) if at least one of the following
statements is true.
limx→a f (x) = ∞
limx→a f (x) = −∞
limx→a+ f (x) = ∞
limx→a+ f (x) = −∞.
Assoc. Prof. Nguyen Ngoc Hai
limx→a− f (x) = ∞
limx→a− f (x) = −∞
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.1
INFINITE LIMITS
Example 9.1
Find the vertical asymptotes of
(a) f (x) = ln x
Assoc. Prof. Nguyen Ngoc Hai
(b) g (x) = tan x.
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
Now we investigate limits at infinity, where x
becomes arbitrarily large, positive or negative.
Definition 9.3
Let f be a function defined on some interval (a, ∞).
Then,
lim f (x) = L
x→∞
means that the values of f (x) can be made
arbitrarily close to L by taking x sufficiently large.
Another notation for limx→∞ f (x) = L is
f (x) → L as x → ∞.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
FIGURE 1.36 Limits at ∞
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
Definition 9.4
Let f be a function defined on some interval
(−∞, a). Then,
lim f (x) = L
x→−∞
means that the values of f (x) can be made
arbitrarily close to L by taking x sufficiently large
negative.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
Definition 9.5
The line y = L is called a horizontal asymptote of
the curve y = f (x) if either
lim f (x) = L
x→∞
or
lim f (x) = L.
x→−∞
Most of the Limit Laws given in Section 1.7 also
hold for limits at infinity.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
FIGURE 1.37 limx→−∞ tan−1 x = − π2 , limx→∞ tan−1 x =
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
π
2
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
Example 9.2
Example 9.3
If n is a positive integer, then
1
lim n = 0.
x→±∞ x
Evaluate
20x 2 − 3x
lim
.
x→±∞ 3x 5 − 4x 2 + 5
Example 9.4 Find the horizontal and vertical
asymptotes of the graph of the function
√
2x 2 + 1
f (x) =
.
3x − 5
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.2
FINITE LIMITS AT INFINITY
Theorem 9.1
The limits
1 x
lim 1 +
x→−∞
x
and
lim
x→∞
1 x
1+
x
exist and equal. This value is called the number e .
Thus we have
1
lim (1 + t) t = e
t→0
ln(1 + t)
=1
t→0
t
lim
eu − 1
lim
=1
u→0
u
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.3
INFINITE LIMITS AT INFINITY
The notation
lim f (x) = ∞
x→∞
is used to indicate that the values of f (x) become
large as x becomes large.
Similar meanings are attached to the following
symbols:
lim f (x) = −∞,
x→∞
lim f (x) = ∞,
x→−∞
lim f (x) = −∞.
x→−∞
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.3
INFINITE LIMITS AT INFINITY
Example 9.5
If a > 1 then
lim ax = 0
x→−∞
and
lim ax = ∞.
x→∞
Example 9.6 Calculate
11x + 2
,
(a) limx→±∞ 3
x −1
−4x 3 + 7x
(b) limx→∞ 2
,
2x − 3x − 10
−4x 3 + 7x
(c) limx→−∞ 2
.
2x − 3x − 10
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
1.9 LIMITS INVOLVING INFINITY
1.9.3
INFINITE LIMITS AT INFINITY
Asymptotic Behavior of a Rational Function
The asymptotic behavior of a rational function
depends only on the leading terms of its numerator
and denominator. Suppose an , bm ̸= 0 and
an x n + an−1 x n−1 + · · · + a0
L = lim
.
x→±∞ bm x m + bm−1 x m−1 + · · · + b0
• If m > n, then L = 0.
• If m = n, then L = an /bm .
• If m < n, then L = ±∞, depending on the signs
of numerator and denominator.
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Exercises and Assignments
Text book: J. Stewart, Calculus. Early Transcendentals, 8th
edition.
Pages
19–23
33–36
42–45
66–68
70
Exercises
Assignments
(corrected in class)
(to be submitted)
7, 8, 25, 29, 43, 54, 75 4, 9, 23, 30, 35, 38
53, 57, 62, 67, 78
1, 18, 21
2, 4, 10, 14, 17, 20, 22
3, 7, 24, 32, 37, 41
5, 23, 42, 48, 59, 61, 63
5, 6, 18, 21, 24, 30
7, 8, 20, 23, 26
19, 23
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity
Exercises and Assignments
Text book: J. Stewart, Calculus. Early Transcendentals, 8th
edition.
Pages
Exercises
Assignments
(corrected in class)
(to be submitted)
92–94 5, 11
4, 7, 9, 15, 54
102–104 2, 31, 36, 37,
8, 12, 26, 32, 38, 39
52, 53, 55, 65
124–126 3, 17, 35, 41, 54
4, 6, 9, 19, 21, 22, 33, 36,
42, 44, 45, 46, 53, 58, 72
137–140 2, 4, 19, 55, 67
3, 20, 27, 48, 57
Assoc. Prof. Nguyen Ngoc Hai
CALCULUS I Chapter 1: Functions, Limits and Continuity