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Read aloud Readability settings Prev Up Next \(\newcommand{\Z}{\mathbb{Z}}
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Activity 2.7 Non-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.
\begin{equation*}
\lim\limits_{x \to 2}\fe{k}{x} = \infty, \lim\limits_{x \to 2^{-}}\fe{k}{x} = \infty,\text{ and }\lim\limits_{x \to 2^{+}}\fe{k}{x} = \infty\text{.}
\end{equation*}
\begin{equation*}
\lim\limits_{t \to 4}\fe{w}{t} = -\infty, \lim\limits_{t \to 4^{-}}\fe{w}{t} = -\infty,\text{ and }\lim\limits_{t \to 4^{+}}\fe{w}{t} = -\infty\text{.}
\end{equation*}
\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 \(\lim\limits_{x \to -3}\fe{T}{x}\text{.}\)
Figure 2.7.1. \(y = \fe{k}{x}\)
Figure 2.7.2. \(y = \fe{w}{t}\)
Figure 2.7.3. \(y = \fe{T}{x}\)
Exercises Exercises
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 \to 3^{-}}\fe{f}{x} \amp= -\infty\\
\lim\limits_{x \to \infty}\fe{f}{x} \amp= 0\\
\lim\limits_{x \to 3^{+}}\fe{f}{x} \amp= \infty\\
\lim\limits_{x \to -\infty}\fe{f}{x} \amp= 2
\end{align*}
Figure 2.7.4. \(y = \fe{f}{x}\)
