# Lesson Explainer: Evaluating Limits Using Algebraic Techniques Mathematics • Higher Education

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

Find .

### Answer

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 .

### Answer

Substitution gives and we note the common factor . Factoring the 8 in the numerator, reveals powers of 2: and so this is .

Now using the fact that

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