In each of the next two examples, the value of the limit is the same as the value of the function at the point it approaches. This is typically the case for any polynomial.

For functions that are not polynomials, a table i often in order to evaluate the limit. In each of the following two examples, the output of the function grow more positive or more negative. This means the different limits do not exist.

However, a very similar looking fraction may also lead to a limit that does exist. In the next two examples, the limits do exist even though the functions are undefined at the point the x value is approaching.

In the next several examples, a graph is used to evaluate one-sided limits from the right and left, the corresponding two-sided limit, and the value of the function. The key is to find the y values as the x values approach some value from the left and the right…if those match, the two sided limit is the y value that it matches at.

Does anyone see any mistakes in the examples above?

You can recognize the limits by what happens when you substitute the value x approaches into the expression. If it gives 0/0, there is algebra that you can do to find the exact value of the limit.

In the first two examples, the expression may be factored and simplified…then you can substitute the value for x.

In the next two examples, the fractions in the numerator must be combined before the fraction may be simplified.

The next two examples are designed to throw you off. When you substitute the value into the expression, you do not get 0/0. This means you need to use a table or graph to get the limit.

The next two examples show how to rationalize the numerator to do a limit.

This example may be done two different ways as the next two boards demonstrate.

One of the problems on the homework gave you three points on a line graph, (1905, 1024), (1955, 240), (2005, 1141). In these ordered pairs, the x value is the year and the y value is the number of immigrants (in thousands) to a large country.

Find the average rate of change in immigration from 1905 to 1955 in immigrants per year.

Find the average rate of change in immigration from 1955 to 2005 in immigrants per year.

Find the average rate of change in immigration from 1905 to 2005 in immigrants per year.

This problem illustrates the two ways that you can work in the “thousands” in the data to give immigrants per year instead of thousands of immigrants per year.