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Section 1.3 Discovering More Limits
For some applications we need specialized limits. The next activity illustrates how to modify the first definition to handle these cases.
Like the previous activity about limits consider
Figure 1.3.1 which shows examples of limits to infinity (or negative infinity) that are defined, and
Figure 1.3.2 which shows examples of limits to infinity (or negative infinity) that are not defined.
\begin{equation*}
\lim_{t \to \infty} f(t)
\end{equation*}
\begin{equation*}
\lim_{y \to \infty} g(y)
\end{equation*}
\begin{equation*}
\lim_{x \to \infty} h(x)
\end{equation*}
\begin{equation*}
\lim_{t \to -\infty} f(t)
\end{equation*}
\begin{equation*}
\lim_{y \to -\infty} g(y)
\end{equation*}
\begin{equation*}
\lim_{x \to -\infty} h(x)
\end{equation*}
\begin{equation*}
\lim_{t \to \infty} f(t)
\end{equation*}
\begin{equation*}
\lim_{y \to \infty} g(y)
\end{equation*}
\begin{equation*}
\lim_{y \to -\infty} g(y)
\end{equation*}
Diagram Exploration Keyboard Controls
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Figure 1.3.1. Each of these limits are defined.
\begin{equation*}
\lim_{x \to \infty} h(x)
\end{equation*}
\begin{equation*}
\lim_{\alpha \to \infty} h(\alpha)
\end{equation*}
\begin{equation*}
\lim_{\alpha \to -\infty} h(\alpha)
\end{equation*}
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Figure 1.3.2. None of these limits are defined.
Activity 1.3.3 . Describe Definining Traits of More Limits.
The goal of this activity is to determine traits of functions where a limit to infinity (or negative infinity) is defined by considering their graphs and to characterize conditions when the limit is not defined.
(a) For each limit given in
Figure 1.3.4 determine if the limit is defined or not by finding a matching example in
Figure 1.3.1 for limits that are defined or in
Figure 1.3.2 for limits that are not defined.
(b) Using the examples in
Figure 1.3.1 ,
Figure 1.3.2 , and
Figure 1.3.4 find traits of the graphs when the limit to infinity (or negative infinity) is defined that are not present in the graphs when the limit to infinity (or negative infinity) is not defined. Likewise find traits of the graphs when the limit to infinity (or negative infinity) is not defined that are not present in the graphs when the limit to infinity (or negative infinity) is defined.
\begin{equation*}
\lim_{x \to \infty} f(x)
\end{equation*}
\begin{equation*}
\lim_{y \to \infty} g(y)
\end{equation*}
\begin{equation*}
\lim_{z \to \infty} h(z)
\end{equation*}
\begin{equation*}
\lim_{x \to -\infty} f(x)
\end{equation*}
\begin{equation*}
\lim_{y \to -\infty} g(y)
\end{equation*}
\begin{equation*}
\lim_{z \to -\infty} h(z)
\end{equation*}
Diagram Exploration Keyboard Controls
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Action
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B
Activate menu driven exploration
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Explore next lower level
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Explore next upper level
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Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
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Figure 1.3.4. Determine which of these limits is defined.
Checkpoint 1.3.5 .
Limits to infinity and to negative infinity require their own definitions.
What parts of the definition need to change?
To what do those parts change?
If we treat two limits as distinct because they have different definitions, how many distinct types of limits have we defined?
