By Rick Stoll |
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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
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
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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