Limits of Polynomial and Rational Functions

By now you have probably noticed that, in each of the previous examples, it has been the case that \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) = f \left(\right. a \left.\right) .\) This is not always true, but it does hold for all polynomials for any choice of a and for all rational functions at all values of a for which the rational function is defined.

Theorem 2.6

Limits of Polynomial and Rational Functions

Let \(p \left(\right. x \left.\right)\) and \(q \left(\right. x \left.\right)\) be polynomial functions. Let a be a real number. Then,

\[\underset{x \rightarrow a}{\text{lim}} p \left(\right. x \left.\right) = p \left(\right. a \left.\right)\]

\[\underset{x \rightarrow a}{\text{lim}} \frac{p \left(\right. x \left.\right)}{q \left(\right. x \left.\right)} = \frac{p \left(\right. a \left.\right)}{q \left(\right. a \left.\right)} \text{when} q \left(\right. a \left.\right) \neq 0 .\]

To see that this theorem holds, consider the polynomial \(p \left(\right. x \left.\right) = c_{n} x^{n} + c_{n - 1} x^{n - 1} + \hdots + c_{1} x + c_{0} .\) By applying the sum, constant multiple, and power laws, we end up with

\[\begin{aligned} \underset{x \rightarrow a}{\text{lim}} p \left(\right. x \left.\right) & = \underset{x \rightarrow a}{\text{lim}} \left(\right. c_{n} x^{n} + c_{n - 1} x^{n - 1} + \hdots + c_{1} x + c_{0} \left.\right) \\ & = c_{n} \left(\left(\right. \underset{x \rightarrow a}{\text{lim}} x \left.\right)\right)^{n} + c_{n - 1} \left(\left(\right. \underset{x \rightarrow a}{\text{lim}} x \left.\right)\right)^{n - 1} + \hdots + c_{1} \left(\right. \underset{x \rightarrow a}{\text{lim}} x \left.\right) + \underset{x \rightarrow a}{\text{lim}} c_{0} \\ & = c_{n} a^{n} + c_{n - 1} a^{n - 1} + \hdots + c_{1} a + c_{0} \\ & = p \left(\right. a \left.\right) . \end{aligned}\]

It now follows from the quotient law that if \(p \left(\right. x \left.\right)\) and \(q \left(\right. x \left.\right)\) are polynomials for which \(q \left(\right. a \left.\right) \neq 0 ,\) then

\[\underset{x \rightarrow a}{\text{lim}} \frac{p \left(\right. x \left.\right)}{q \left(\right. x \left.\right)} = \frac{p \left(\right. a \left.\right)}{q \left(\right. a \left.\right)} .\]

Example 2.16 applies this result.

Example 2.16

Evaluating a Limit of a Rational Function

Evaluate the \(\underset{x \rightarrow 3}{\text{lim}} \frac{2 x^{2} - 3 x + 1}{5 x + 4} .\)

Solution

Since 3 is in the domain of the rational function \(f \left(\right. x \left.\right) = \frac{2 x^{2} - 3 x + 1}{5 x + 4} ,\) we can calculate the limit by substituting 3 for x into the function. Thus,

\[\underset{x \rightarrow 3}{\text{lim}} \frac{2 x^{2} - 3 x + 1}{5 x + 4} = \frac{10}{19} .\]

Checkpoint 2.12

Evaluate \(\underset{x \rightarrow −2}{\text{lim}} \left(\right. 3 x^{3} - 2 x + 7 \left.\right) .\)

This lesson is part of:

Limits

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