Essential Concepts
- If [latex]c[/latex] is a critical point of [latex]f[/latex] and [latex]f^{\prime}(x)>0[/latex] for [latex]x
c[/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]x
0[/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]
Candela Citations
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- Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/details/books/calculus-volume-1. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction