## Limit of a function

If a function is defined by , then the values of depend on the values of .

As the values of are getting closer and closer to also gets arbitrarily close to a single value , say .

Mathematically this limit is denoted by:

This is read as “the limit of as approaches , is ”. This is illustrated diagrammatically in fig 1.0 below:

## Finding the limit of a Polynomial function

If is a polynomial function and is any real number, then the limit of as approaches is the same as the function value at ‘’. That is:

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Use the graph of y= figure 1.1 to evaluate:

Solution

## Finding the limit of a rational function

If is a rational function given by , for any real number r,

NB: If yields , then we must further simplify before we evaluate.

Example 5

Solution

Since the denominator is not zero when , we can substitute directly into the function:

Example 6

Solution

Again the denominator is not zero when , we can substitute directly into the function:

Example 7

Solution

Direct substitution yields, . This is an implication that we need to simplify the given rational function before we evaluate.

**Example 8:**

Evaluate

**Solution**

Again direct substitution yields, . Therefore we need to simplify the given rational function before we evaluate.

Recall from binomial expansion that . We can rearrange to get an expression for the numerator, .

That is:

Hence,

Example 9

Solution

Once again, direct substitution yields, . Therefore we need to rationalize the denominator of the given rational function before we evaluate.