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Calculus I: Activities

Mark A. Fitch

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.
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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.
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\begin{equation*} \lim_{t \to \infty} f(t) \end{equation*}
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\begin{equation*} \lim_{y \to \infty} g(y) \end{equation*}
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\begin{equation*} \lim_{x \to \infty} h(x) \end{equation*}
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TL
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TC
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TR
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\begin{equation*} \lim_{t \to -\infty} f(t) \end{equation*}
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\begin{equation*} \lim_{y \to -\infty} g(y) \end{equation*}
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\begin{equation*} \lim_{x \to -\infty} h(x) \end{equation*}
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CL
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CC
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CR
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\begin{equation*} \lim_{t \to \infty} f(t) \end{equation*}
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\begin{equation*} \lim_{y \to \infty} g(y) \end{equation*}
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\begin{equation*} \lim_{x \to -\infty} h(x) \end{equation*}
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BL
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BC
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BR
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Figure 1.3.1. Each of these limits are defined.
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\begin{equation*} \lim_{x \to \infty} h(x) \end{equation*}
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\begin{equation*} \lim_{\alpha \to \infty} h(\alpha) \end{equation*}
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\begin{equation*} \lim_{\alpha \to -\infty} h(\alpha) \end{equation*}
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TL
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TC
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TR
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Figure 1.3.2. None of these limits is defined.
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Activity 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.
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(b)

Using the examples in Figure 1.3.1, Figure 1.3.2, and Figure 1.3.3 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.
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\begin{equation*} \lim_{x \to \infty} f(x) \end{equation*}
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\begin{equation*} \lim_{y \to \infty} g(y) \end{equation*}
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\begin{equation*} \lim_{z \to \infty} h(z) \end{equation*}
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\begin{equation*} \lim_{x \to -\infty} f(x) \end{equation*}
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\begin{equation*} \lim_{y \to -\infty} g(y) \end{equation*}
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\begin{equation*} \lim_{z \to -\infty} h(z) \end{equation*}
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Figure 1.3.3. Determine which of these limits defined.
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Checkpoint 1.3.4.

Limits to infinity and to negative infinity require their own definitions.
  1. What parts of the definition need to change?
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  2. To what do those parts change?
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  3. If we treat two limits as distinct because they have different definitions, how many distinct types of limits have we defined?
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