Lecture1| 1
Chapter 1
Limit and Continuity of Functions
1.1.
Definition of Limits
Consider the function
We cannot plug , that is is undefined.
Consider several values for close to . From the table, it follows that the value approaches 3 as approaches .
Lecture1| 2 We denote this by
This is read as
the limit of as approaches is ,
is called the limit of as approaches .
Lecture1| 3 Definition. For a function and a number , we say that the limit of as approaches is equal to , written
if the value approaches as approaches . Notation. We write
for approaches, and
for approaches .
Lecture1| 4 EXAMPLE. Consider the limit of
as . (Note, is undefined at !)
From the table, we see that as so
Lecture1| 5 EXAMPLE. Consider the limit of the function
as approaches .
From the graph, we see that
The value does not affect the limit.
Lecture1| 6 For, example, if
we still have
So while
Summary. In finding , it doesn’t matter what is.
Methods to find limits 1. Numerical (tables) 2. Graphical (graphs)
3. Analytical (algebra and calculus).
Lecture1| 7 Definition. For a function and a number , we say that
if either
1. approaches different numbers as approaches from different directions, or 2. does not approach any number as
approaches .
Note. The case 2 can happen for instance, when becomes arbitrary large as .
is oscillatory as . See examples below.
Lecture1| 8 EXAMPLE (Different left and right behaviors).
Show that the limit of
as does not exist.
From the graph, we see that
Lecture1| 9 EXAMPLE (Unbounded behavior).
Discuss the existence of the limit of
as approaches .
From the graph, we find that
Lecture1| 10 In fact, increases without bound as
approaches .
Lecture1| 11 EXAMPLE (Oscillatory behavior).
Discuss the existence of the limit
From the graph, we find that
Lecture1| 12 Question Use a calculator to find several values
where
and estimate the limit .
Lecture1| 13 Question For each of the following functions,
discuss the given limit from graphs.
Lecture1| 14
Lecture1| 15
/
Lecture1| 16 Question From the graph of given below
find the following 1.
2.
3.
4.
Lecture1| 17 1.2.
One-sided Limits
Definition. For a function and a number ,
if approaches as approaches and , we call the left-hand limit written
if approaches as approaches and , we call the right-hand limit written
Lecture1| 18 EXAMPLE. Find the limit of as
approaches from the right.
From the graph, we get
Note. is undefined for , so the left- hand limit is undefined.
Lecture1| 19 Question Use the graph to find each of the one-
sided limits
Lecture1| 20
Lecture1| 21
Lecture1| 22 Theorem if and only if
and
Lecture1| 23 EXAMPLE. Discuss the limit at of
whose graph is as shown.
From the graph,
which are not equal. So
Lecture1| 24 EXAMPLE. Discuss the limit for the
function whose graph is as shown.
From the graph,
and the right-hand limit as well. So
Lecture1| 25
1.3.
Limit Laws
Next, we present some limit laws that will be useful in finding limits analytically.
Rule 1 (Limit of constant).
Lecture1| 26 Rule 2 (Limit of ).
Lecture1| 27 Rule 3 (Limit of and 𝒏 ).
and
whenever the right-hand side is defined.
Here , natural numbers.
Lecture1| 28 EXAMPLE. Evaluate the following limits.
1.
2.
3.
4.
Lecture1| 29 Rule 4 (Algebra of limits). Suppose
1. Scalar multiple
2. Sum or difference
3. Product
4. Quotient:
Lecture1| 30
EXAMPLE. Let and .
Find the following limits 1.
2.
3.
4.
Lecture1| 31 EXAMPLE. Find the limit
Lecture1| 32 Rule 5 (Direct substitution). Let be polynomials. Then
and
Polynomials are functions of the form
Lecture1| 33 EXAMPLE. Find the following limits
1.
2.
3.
Lecture1| 34 Rule 6 (Composite).
and
Lecture1| 35 EXAMPLE. Find each of the following limits
1.
2.
3.
