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
βindeterminate formβ .

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

on the set of all
numbers where both functions are defined, which is the set
,

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 .

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:

Example 2: Finding the Limit of Rational Functions by Eliminating Common Factors

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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