Summary of Limits at Infinity and Asymptotes

If $c$ is a critical point of $f$ and $f^{\prime}(x)>0$ for [latex]xc[/latex], then $f$ has a local maximum at $c$ .
If $c$ is a critical point of $f$ and $f^{\prime}(x)<0$ for [latex]x0[/latex] for $x>c$ , then $f$ has a local minimum at $c$ .
For a polynomial function $p(x)=a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ , where $a_n \ne 0$ , the end behavior is determined by the leading term $a_n x^n$ . If $n\ne 0$ , $p(x)$ approaches $\infty$ or $−\infty$ at each end.
For a rational function $f(x)=\frac{p(x)}{q(x)}$ , the end behavior is determined by the relationship between the degree of $p$ and the degree of $q$ . If the degree of $p$ is less than the degree of $q$ , the line $y=0$ is a horizontal asymptote for $f$ . If the degree of $p$ is equal to the degree of $q$ , then the line $y=\frac{a_n}{b_n}$ is a horizontal asymptote, where $a_n$ and $b_n$ are the leading coefficients of $p$ and $q$ , respectively. If the degree of $p$ is greater than the degree of $q$ , then $f$ approaches $\infty$ or $−\infty$ at each end.

Key Equations

Infinite Limits from the Left
$\underset{x\to a^-}{\lim}f(x)=+\infty$
$\underset{x\to a^-}{\lim}f(x)=−\infty$
Infinite Limits from the Right
$\underset{x\to a^+}{\lim}f(x)=+\infty$
$\underset{x\to a^+}{\lim}f(x)=−\infty$
Two-Sided Infinite Limits
$\underset{x\to a}{\lim}f(x)=+\infty: \underset{x\to a^-}{\lim}f(x)=+\infty$ and $\underset{x\to a^+}{\lim}f(x)=+\infty$
$\underset{x\to a}{\lim}f(x)=−\infty: \underset{x\to a^-}{\lim}f(x)=−\infty$ and $\underset{x\to a^+}{\lim}f(x)=−\infty$

end behavior: the behavior of a function as $x\to \infty$ and $x\to −\infty$

horizontal asymptote: if $\underset{x\to \infty }{\lim}f(x)=L$ or $\underset{x\to −\infty }{\lim}f(x)=L$ , then $y=L$ is a horizontal asymptote of $f$

infinite limit at infinity: a function that becomes arbitrarily large as $x$ becomes large

limit at infinity: the limiting value, if it exists, of a function as $x\to \infty$ or $x\to −\infty$

oblique asymptote: the line $y=mx+b$ if $f(x)$ approaches it as $x\to \infty$ or $x\to −\infty$