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 2.7.12.7.3 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 $\lim\limits_{x\to-3}\fe{T}{x}\text{.}$

# 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*}