Learn the rules for finding limits of rational functions as x approaches +infinity or -infinity. This involves comparing the degrees of the polynomials in the numerator and denominator to determine horizontal asymptotes.
What does the limit at infinity, \(\lim_{x \to \infty} f(x) = L\), represent graphically?
It represents a horizontal asymptote for the graph of \(f(x)\). The line \(y=L\) is the horizontal asymptote.
What is the fundamental algebraic step for finding the limit at infinity of a rational function?
Divide every single term in both the numerator and the denominator by the highest power of \(x\) present in the denominator.
Let \(n\) be the degree of the numerator and \(m\) be the degree of the denominator of a rational function. What is the limit as \(x \to \pm\infty\) if \(n < m\)?
The limit is \(0\). The denominator grows faster than the numerator, so the fraction approaches zero. The horizontal asymptote is \(y=0\).
The degree of the numerator (\(2\)) is less than the degree of the denominator (\(3\)). Therefore, the limit is \(0\).
Let \(n\) be the degree of the numerator and \(m\) be the degree of the denominator. What is the limit as \(x \to \pm\infty\) if \(n = m\)?
The limit is the ratio of the leading coefficients of the numerator and denominator. The horizontal asymptote is \(y = \frac{a_n}{b_m}\), where \(a_n\) and \(b_m\) are the leading coefficients.
The degrees of the numerator and denominator are equal (\(3\)). The limit is the ratio of the leading coefficients: \(\frac{4}{8} = \frac{1}{2}\).
Let \(n\) be the degree of the numerator and \(m\) be the degree of the denominator. What is the limit as \(x \to \pm\infty\) if \(n > m\)?
The limit does not exist. The function will approach either positive infinity (\(+\infty\)) or negative infinity (\(-\infty\)). There is no horizontal asymptote.
$$\lim_{x \to \infty} \frac{x^4 - 5}{x^2 + 3x}$$
The degree of the numerator (\(4\)) is greater than the degree of the denominator (\(2\)). As \(x\) becomes a large positive number, both the numerator and denominator are positive, so the limit is \(+\infty\).
How do you determine whether a limit approaches \(+\infty\) or \(-\infty\) when the numerator's degree is larger than the denominator's?
Analyze the 'end behavior model' formed by the leading terms. For \(\frac{a_n x^n}{b_m x^m}\), consider the sign of \(\frac{a_n}{b_m}\) and whether \(x^{n-m}\) becomes positive or negative as \(x \to \infty\) or \(x \to -\infty\).
The degree of the numerator is greater than the denominator. The end behavior is like \(\frac{5x^3}{2x^2} = \frac{5}{2}x\). As \(x \to -\infty\), \(\frac{5}{2}x\) also approaches \(-\infty\). The limit is \(-\infty\).
The effective degree of the numerator is \(\sqrt{x^6} = |x^3|\). As \(x \to -\infty\), \(|x^3| = -x^3\). So the limit is $$\lim_{x \to -\infty} \frac{-3x^3}{2x^3} = -\frac{3}{2}$$
