## 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

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

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