# Explainer: Evaluating Limits Using Algebraic Techniques

In this explainer, we will learn how to use algebraic techniques like factorization to evaluate limits.

Specifically, we will cover the case where in seeking the limit ,

• the function is a rational function: , with and polynomials in ,
• substitution fails because the quotient gives the “indeterminate form” .

An example is , where and give and

Since both and are polynomials, means that is a factor of each polynomial. So we can rewrite and divide these factors out:

Now with the new function we see that

1. on the set of all numbers where both functions are defined, which is the set ,
2. the function is defined at (unlike ).

From (2), we can calculate the limit of at by substitution:

Next, we use the following general principle.

### Limits and Almost Equal Functions

Suppose that

• functions and are equal at all points of an interval except ,
• has the limit as tends to .

Then, also.

Now, fact (1) allows us to conclude that

### Example 1: Finding the Limit of Rational Functions by Eliminating Common Factors

Find .

Since and also, we need to factor out

So

The following factorization extends the difference of squares : and then and, more generally, which can be written as from which we deduce that since the right-hand side has terms.

This shortens potentially lengthy factorizations and has the following useful corollary:

Find .