Limits
Some Basic Properties
To start out with, let us note that the limit, when it exists, is unique. That is why we
say "the limit", not "a limit". This property translates formally into:
Most of the examples studied before used the definition of the limit. But in general it
is tedious to find the given the . The following properties help circumvent this.
Theorem. Let f(x) and g(x) be two functions. Assume that
Then
(1)
;
(2)
, where
is an arbitrary number;
(3)
.
These properties are very helpful. For example, it is easy to check that
for any real number a. So Property (3) repeated implies
and Property (2) implies
These limits combined with Property (1) give
for any polynomial function
.
The next natural question then is to ask what happens to quotients of functions. The
following result answers this question:
Theorem. Let f(x) and g(x) be two functions. Assume that
Then
provided
.
This implies immediately the following:
where P(x) and Q(x) are two polynomial functions with
Example. Assume that
Find the limit
.
Answer. Note that we cannot apply the result about limits of quotients directly, since
the limit of the denominator is zero. The following manipulations allow to circumvent
this problem. We have
Using the above properties we get
and
Hence
which gives the limit
Exercise 1. Find the limit
At first glance, you might be led to believe that this limit does not exist, since the
denominator is equal to 0 when x=1, But then you will notice that the numerator is
also equal to 0 when x=1. In this case, since we are dealing with polynomials, we can
find a factor (x-1) in the numerator. Indeed, we have x2 - 3x +2 = (x-1)(x-2). So, for
,
which implies
Exercise 2. Evaluate
We have
Since
Also we have
Since
and
, then
then
Let us continue to list some basic properties of limits.
Theorem. Let f(x) be a positive function, i.e.
. Assume that
Then
This is actually a special case of the following general result about the composition of
two functions:
Theorem. Let f(x) and g(x) be two functions. Assume that
Then
Example. Geometric considerations imply
Since
for x close to 0, then we have
which implies
Using the trigonometric identities
we obtain