## 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 [Back] [Next]
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Mohamed A. Khamsi
Helmut Knaust