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Activity2.7Non-existent Limits

Many limit values do not exist. Sometimes the non-existence is caused by the function value either increasing without bound or decreasing without bound. In these special cases we use the symbols \(\infty\) and \(-\infty\) to communicate the non-existence of the limits. Figures 2.7.1–2.7.3 can be used to illustrate some ways in which we communicate the non-existence of these types of limits.

  • In Figure 2.7.1 we could (correctly) write

    \begin{equation*} \lim\limits_{x\to2}\fe{k}{x}=\infty, \lim\limits_{x\to2^{-}}\fe{k}{x}=\infty,\text{ and }\lim\limits_{x\to2^{+}}\fe{k}{x}=\infty\text{.} \end{equation*}

  • In Figure 2.7.2 we could (correctly) write

    \begin{equation*} \lim\limits_{t\to4}\fe{w}{t}=-\infty, \lim\limits_{t\to4^{-}}\fe{w}{t}=-\infty,\text{ and }\lim\limits_{t\to4^{+}}\fe{w}{t}=-\infty\text{.} \end{equation*}

  • In Figure 2.7.3 we could (correctly) write

    \begin{equation*} \lim\limits_{x\to-3^{-}}\fe{T}{x}=\infty\text{ and }\lim\limits_{x\to-3^{+}}\fe{T}{x}=-\infty\text{.} \end{equation*}

    There is no shorthand way of communicating the non-existence of the two-sided limit \(\lim\limits_{x\to-3}\fe{T}{x}\text{.}\)

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Figure2.7.1\(y=\fe{k}{x}\)
Figure2.7.2\(y=\fe{w}{t}\)
Figure2.7.3\(y=\fe{T}{x}\)

Subsection2.7.1Exercises

1

In the plane provided, draw the graph of a single function, \(f\text{,}\) that satisfies each of the following limit statements. Make sure that you draw the necessary asymptotes and that you label each asymptote with its equation.

\begin{align*} \lim\limits_{x\to3^{-}}\fe{f}{x}&=-\infty\\ \lim\limits_{x\to\infty}\fe{f}{x}&=0\\ \lim\limits_{x\to3^{+}}\fe{f}{x}&=\infty\\ \lim\limits_{x\to-\infty}\fe{f}{x}&=2 \end{align*}

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Figure2.7.4\(y=\fe{f}{x}\)