Math 22 Limits

From Math Wiki
Jump to navigation Jump to search

The Limit of a Function

 Definition of the Limit of a Function
 If  becomes arbitrarily close to a single number  as  approaches  from either side, then
 which is read as "the limit of  as  approaches  is 

Note: Many times the limit of as approaches is simply , so limit can be evaluate by direct substitution as

Properties of Limits

Let and be real numbers, let be a positive integer, and let and be functions with the following limits and . Then

1. Scalar multiple:

2. Sum or difference:

3. Product:

4. Quotient:

5. Power:

6. Radical:

Techniques for Evaluating Limits

1. Direct Substitution: Direct Substitution can be used to find the limit of a Polynomial Function.

Example: Evaluate

2. Dividing Out Technique: When direct substitution fails and numerator or/and denominator can be factored.

Example: Evaluate . Now we can use direct substitution to get the answer.

3. Rationalizing (Using Conjugate): When direct substitution fails and either numerator or denominator has a square root. In this case, we can try to multiply both numerator and denominator by the conjugate.

Example: Evaluate . Now we can use direct substitution to get the answer

One-Sided Limits and Unbounded Function

 when a function approaches a different value from the left of  than it approaches from the right of , the limit does not exists. However, this type of behavior can be described more concisely with 
 the concept of a one-sided limit. We denote

One-sided Limit is related to unbounded function.

In some case, the limit of can be increase/decrease without bound as approaches . We can write

Now, consider . By direct substitution, it is of the form , so the answer will be either or . In order to find the limit, we must consider the limit from both side ( and ).

When , so , hence . Therefore,

When , so , hence . Therefore,

Notice: .

So, does not exists

Return to Topics Page

This page were made by Tri Phan