Summary of Limits at Infinity and Asymptotes

Essential Concepts

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

Key Equations

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

Glossary

end behavior
the behavior of a function as [latex]x\to \infty[/latex] and [latex]x\to −\infty[/latex]
horizontal asymptote
if [latex]\underset{x\to \infty }{\lim}f(x)=L[/latex] or [latex]\underset{x\to −\infty }{\lim}f(x)=L[/latex], then [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex]
infinite limit at infinity
a function that becomes arbitrarily large as [latex]x[/latex] becomes large
limit at infinity
the limiting value, if it exists, of a function as [latex]x\to \infty[/latex] or [latex]x\to −\infty[/latex]
oblique asymptote
the line [latex]y=mx+b[/latex] if [latex]f(x)[/latex] approaches it as [latex]x\to \infty[/latex] or [latex]x\to −\infty[/latex]