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
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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