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
These properties are very helpful. For example, it is easy to
check that
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
This implies immediately the following:
Example. Assume that
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
Let us continue to list some basic properties of limits.
Theorem. Let f(x) be a positive function, i.e.
.
Assume that
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
Example. Geometric considerations imply
Using the trigonometric identities
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Mohamed A. Khamsi