Limits of Polynomials and Rational Functions

This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

# Limits of Polynomials and Rational Functions

Before we look at some theorems regarding the limits of polynomials and rational functions, we should first formally define what each is.

 Definition: A function in the form $p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$ where $a_0, a_1, ..., a_n \in \mathbb{R}$ is said to be a polynomial of degree $n$. If $p(x)$ and $q(x)$ are polynomials, then the function $r(x) = \frac{p(x)}{q(x)}$, $q(x) \neq 0$ is said to be a rational function.

For example, the $p : \mathbb{R} \to \mathbb{R}$ defined by $p(x) = 3 + 2x^2 + 4x^4$ is a polynomial, and the function $r : \mathbb{R} \setminus \{ 1 \} \to \mathbb{R}$ defined by $r(x) = \frac{2x + 3x^2}{1 - x}$ is a rational function.

We will now look at some theorems regarding the limits of these functions.

 Theorem 1: If $p(x) = a_0 + a_1x + ... + a_nx^n$, $a_0, a_1, ..., a_n \in \mathbb{R}$ is a polynomial function. Then the limit at $x = c$ exists and $\lim_{x \to c} p(x) = p(c)$.
• Proof: Let $p(x) = a_0 + a_1x + ... + a_nx^n$ where $a_0, a_1, ..., a_n \in \mathbb{R}$ be a polynomial function. Then we have that:
(1)
\begin{align} \lim_{x \to c} p(x) = \lim_{x \to c} a_0 + a_1x + ... + a_nx^n \\ \lim_{x \to c} p(x) = \lim_{x \to c} \left ( a_0 \right ) + \lim_{x \to c} \left ( a_1x \right )+ ... + \lim_{x \to c} \left ( a_nx^n \right ) \\ \lim_{x \to c} p(x) = a_0 \lim_{x \to c} \left ( 1 \right ) + a_1 \lim_{x \to c} \left ( x \right ) + ... + a_n \lim_{x \to c} \left ( x^n \right ) \end{align}
• Now recall that $\lim_{x \to c} 1 = 1$ and $\lim_{x \to c} x = c$. Furthermore, from the Operations on Functions and Their Limits page, recall that since $\lim_{x \to c} x = c$, then $\lim_{x \to c} x^2 = \lim_{x \to c} x \cdot \lim_{x \to c} x = c^2$, …, $\lim_{x \to c} x^n = c^n$ (this can be proven by induction), and so:
(2)
\begin{align} \quad \lim_{x \to c} p(x) = a_0 \cdot 1 + a_1 \cdot c + ... + a_n \cdot c^n \\ \lim_{x \to c} p(x) = a_0 + a_1c + ... + a_nc^n \\ \lim_{x \to c} p(x) = p(c) \quad \blacksquare \end{align}
 Theorem 2: If $r(x) = \frac{p(x)}{q(x)}$ is a rational function where $q(c) \neq 0$, then the limit at $x = c$ exists and $\lim_{x \to c} r(x) = \frac{p(c)}{q(c)}$.
• Let $r(x) = \frac{p(x)}{q(x)}$ be a rational function. From theorem 1, since $p(x)$ and $q(x)$ are polynomials, we have that $\lim_{x \to c} p(x) = p(c)$ and $\lim_{x \to c} q(x) = q(c)$. Therefore by the Quotient Law for limits, $\lim_{x \to c} r(x) = \lim_{x \to c} \frac{p(x)}{q(x)} = \frac{p(c)}{q(c)}$, which is valid since $q(c) \neq 0$. $\blacksquare$