Infinite Limits and Limits at Infinity

Although infinite limits and limits at infinity are different, they both describe behavior of functions when either the inputs or outputs increase or decrease "without bound". A third scenario exists for infinite limits at infinity. In this case both the inputs and outputs increase or decrease without bound. In all of these situations even though ∞ is not a real number, it is useful for describing the without bound concept. View the book below to learn about infinite limits, a limit at infinity, and infinite limits at infinity.

In a problem such as equation image indicator $LaTeX: \lim _{x\to \infty }2x^2-x$ we cannot apply the Limit Laws because $LaTeX: \infty -\infty$ - cannot be defined and is not a number $LaTeX: \left(\infty-\infty\ne0\right)$ . Instead, we can write $LaTeX: \lim _{x\to \infty }\left(2x^2-x\right)=\lim _{x\to \infty }\left(2x-1\right)=\infty$ equation image indicator since both x and 2x - 1 become arbitrarily large and thus their product does as well.

Asymptotic Behavior

A function displays asymptotic behavior when it becomes arbitrarily close to an asymptote as the function increases or decreases without bound. Asymptotes come in several varieties: vertical, horizontal, and end behavior (slant). Typically, vertical asymptotes reveal sudden spikes or other anomalies; horizontal and end behavior asymptotes reflect long-term behavior.

Vertical Asymptotes

The line x = a is a vertical asymptote if at least one of the following statements is true:

$LaTeX: \lim _{x\to a^+ }f\left(x\right)=\infty$ equation image indicator $LaTeX: \lim _{x\to a^-}f\left(x\right)=\infty$ $LaTeX: \lim _{x\to a^+ }f\left(x\right)=-\infty$ $LaTeX: \lim _{x\to a^-}f\left(x\right)=-\infty$

Click HERE to view the presentation below to examine how the vertical asymptotes in the graphs satisfy the given limit statements.

More formally, let f and g be continuous on an open interval containing a. If $LaTeX: f\left(a\right)\ne0$ and g(a) = 0, and there exists an open interval containing a, such that for all equation image indicator $LaTeX: x\ne a$ ,

equation image indicator $LaTeX: g\left(x\right)\ne0$ , then the graph of $LaTeX: h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ has a vertical asymptote at x = a. Informally, given a rational function, the vertical asymptotes are located at zeros of the denominator, after canceling any common factors. Note that this method does not apply for other function types.

Horizontal Asymptotes

The line y = L is a horizontal asymptote of the curve y = f(x) if either equation image indicator $LaTeX: \lim _{x\to \infty }f\left(x\right)=L \: or \: \lim _{x\to -\infty }f\left(x\right)=L$ . Unlike vertical asymptotes, horizontal asymptotes may be crossed multiple times by the graph of the function. It is the behavior of f as x approaches $LaTeX: \pm \infty$ that is of importance. Click HERE to view the presentation below to observe graphs of curves approaching a line y = L and displaying one or more horizontal asymptotes

End Behavior (Slant) Asymptotes

End behavior or slant asymptotes arise when graphing a rational function equation image indicator $LaTeX: h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ , where the degree of the numerator is greater than the degree of the denominator. Unlike horizontal and vertical asymptotes, an end behavior asymptote need not be a line. Click HERE to observe the end behavior (slant) asymptotes in each of the graphs.

More formally, let equation image indicator $LaTeX: h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ be a rational function and q(x) and r(x) be the quotient and remainder when f(x) is divided by g(x), i.e., $LaTeX: h\left(x\right)=q\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$ with the degree of r less than the degree of g. Then the graph of q is the end behavior asymptote of h. View the presentation on limits at infinity.

Infinite Limits and Limits at Infinity Practice

Determine which type(s) of asymptotic behavior occur(s) for the given function. Drag the equation to the correct category.

Infinite Limits and Limits at Infinity: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of infinite limits, limits at infinity, and asymptotic behavior.

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