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
- 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
- functions and are equal at all points of an interval except ,
- has the limit as tends to .
Now, fact (1) allows us to conclude that
Example 1: Finding the Limit of Rational Functions by Eliminating Common Factors
Since and also, we need to factor out
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
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