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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.

  • 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 \(\lim\limits_{x\to-3}\fe{T}{x}\text{.}\)

a plot of a function; the curve comes in from the left low on the graph at about y=-6.5; it is flat for a while, but when it reaches the x-axis it begins to curve up steeply; there is a dashed vertical line at x=2, and the curve bends upward until it is almost parallel with this vertical line; the curve is quite close to this vertical line (but still to its left) when the curve leaves the top of the graph; the curve picks up again on the other side of the vertical line, coming in downward from the top of the graph a little to the right of the vertical line; it moves almost straight downward and slightly to the right for a while, crossing below the x-axis; around when it reaches the point (3,-5.5), it flattens as it continues to move rightward
epspdfpngsvgtex
a plot of a function; the curve comes in from the left high on the graph at about y=6.5; it is flat for a while, but when it reaches about x=2 it begins to curve down steeply; there is a dashed vertical line at x=4, and the curve bends downward until it is almost parallel with this vertical line; the curve is quite close to this vertical line (but still to its left) when the curve leaves the bottom of the graph; the curve picks up again on the other side of the vertical line, coming in upward from the bottom of the graph a little to the right of the vertical line; it moves almost straight upward and slightly to the right for a while, crossing above the x-axis; around when it reaches the point (5,5.5), it flattens as it continues to move rightward
epspdfpngsvgtex
a plot of a function; the curve comes in from the left low on the graph at about y=-6.5; it is flat for a short while, but when it reaches about x=-5, it begins to curve up steeply; there is a dashed vertical line at x=-3, and the curve bends upward until it is almost parallel with this vertical line; the curve is quite close to this vertical line (but still to its left) when the curve leaves the top of the graph; the curve picks up again on the other side of the vertical line, coming in upward from the bottom of the graph a little to the right of the vertical line; it moves almost straight upward and slightly to the right for a while, crossing above the x-axis; around when it reaches the point (-2,5.5), it flattens as it continues to move rightward
epspdfpngsvgtex
Figure 2.7.1. \(y=\fe{k}{x}\)
Figure 2.7.2. \(y=\fe{w}{t}\)
Figure 2.7.3. \(y=\fe{T}{x}\)

Subsection 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\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*}
a blank Cartesian grid, with x-axis covering the interval from -7 to 7, and y-axis covering the interval from -7 to 7
epspdfpngsvgtex
Figure 2.7.4. \(y=\fe{f}{x}\)
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