In this section you'll learn to find the limit at infinity of polynomial functions and rational functions. We'll see that if the numerator and denominator of a rational function are polynomials, we can determine a limit at infinity quite easily. When the degree of the numerator is greater than the degree of the denominator, the limit goes to infinity. When the degree of the numerator is less than the degree of the denominator, the limit goes to zero. And when the degree of the numerator is equal to the degree of the denominator, the limit's numerator is the leading coefficient of the function's numerator, and the limit's denominator is the leading coefficient of the function's denominator. After mastering this concept, we'll look at rational functions where the denominator is the square root of a polynomial. At this time, we'll see some other functions whose limit can be determined only after a bit of algebra.

$$\begin{align}
& f(x) = \dfrac{3x - 2}{\sqrt{2x^2 + 1}} \\[1em]
& \lim_{x \to \infty} f(x) = \mathop{?} \\[1em]
& \lim_{x \to -\infty} f(x) = \mathop{?} \\[1em]
\end{align}$$

\(\displaystyle \lim_{x \to \infty} \dfrac{2 - x}{\sqrt{7 + 6x^2}}\)

\(\displaystyle \lim_{x \to \infty} \dfrac{2x^2 - 3}{4x^2 + 5}\)

\(\displaystyle \lim_{x \to \infty} \dfrac{1 - e^x}{1 + e^x}\)

- \(\begin{align}
& f(x) = \dfrac{3x^2 - 8x + 12}{5x^3 + 4x^2 - x - 2} \\[1em]
& \lim_{x \to \infty} f(x) = \mathop{?} \\[1em]
& \lim_{x \to -\infty} f(x) = \mathop{?} \\[1.5em]
\end{align}\)
- \(\begin{align}
& g(x) = \dfrac{3x^5 - 8x^3 + 12}{5x^5 + 4x^2 - x - 2} \\[1em]
& \lim_{x \to \infty} g(x) = \mathop{?} \\[1em]
& \lim_{x \to -\infty} g(x) = \mathop{?} \\[1.5em]
\end{align}\)
- \(\begin{align}
& h(x) = \dfrac{4x^3 - 2x^2 + 4x - 7}{2x^2 + 5x + 9} \\[1em]
& \lim_{x \to \infty} h(x) = \mathop{?} \\[1em]
& \lim_{x \to -\infty} h(x) = \mathop{?} \\[1.5em]
\end{align}\)