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 $q(c) \neq 0$. $\blacksquare$