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Calculus I: Activities

Mark A. Fitch

Section 1.2 Discovering Limits

A major theme of this course is learning properties of curves that make them nice (at least in some context). The first of these properties is a limit being defined at a point, or, even better, the limit being defined on an interval. This just means that the limit is defined at every point in the interval.
In this section we are describing limits visually. We will work on a symbolic definition for use in calculations in other sections. For now we will be satisfied by being able to give visual descriptions of what functions (graphs) look like at a point or on an interval where the limit is defined.
This section is entitled discovering limits. Why is it discover rather than learn? This is to mimic mathematics in life. In life people typically notice a recurring trait in something (make an observation). They take time to describe the trait carefully likely using many examples. Through this process of constructing a description, they create a definition. This is to say that definitions are not printed on the universe; they are simply the result of people describing a phenomenon in a way that enables them to work with it.
In this section we will practice this definition constructing process. Because the examples below have been curated, we will end up noticing some important traits of functions that led over a couple hundred years to the modern definitions for limits.
Our first step in understanding when a limit is defined is to consider Figure 1.2.1 which shows examples of points on curves where the limit is defined, and Figure 1.2.2 which shows examples of points on curves where the limit is not defined. For now just ask yourself what you notice. Write this down.
\begin{equation*} \lim_{t \to 6} f(t) \end{equation*}
\begin{equation*} \lim_{y \to \frac{3\pi}{4}} g(y) \end{equation*}
\begin{equation*} \lim_{x \to 3.5} h(x) \end{equation*}
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\begin{equation*} \lim_{x \to 0} f(x) \end{equation*}
\begin{equation*} \lim_{x \to 1} f(x) \end{equation*}
\begin{equation*} \lim_{t \to 4} h(t) \end{equation*}
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\begin{equation*} \lim_{\gamma \to 0} f(\gamma) \end{equation*}
\begin{equation*} \lim_{\gamma \to 2} f(\gamma) \end{equation*}
\begin{equation*} \lim_{\alpha \to 2} h(\alpha) \end{equation*}
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Figure 1.2.1. Each of these limits is defined.
\begin{equation*} \lim_{x \to 2} h(x) \end{equation*}
\begin{equation*} \lim_{\alpha \to 1} h(\alpha) \end{equation*}
\begin{equation*} \lim_{t \to 0} o(t) \end{equation*}
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Figure 1.2.2. None of these limits is defined.

Activity 1.2.1. Describe Defining Traits of Limits.

The goal of this activity is to determine traits of functions where a limit is defined by considering their graphs and to characterize conditions where the limit is not defined.
Remember definitions are not correct or incorrect; rather they are useful or not. In this activity you are not looking for the correct answer; rather you are looking for patterns that can be turned into a definition. Every pattern you find could be useful in some context. We may end up with more than one, useful definition.

(a)

For each limit given in Figure 1.2.3 determine if the limit is defined or not by finding a matching example in Figure 1.2.1 for limits that are defined or in Figure 1.2.2 for limits that are not defined.

(b)

Using the examples in Figure 1.2.1, Figure 1.2.2, and Figure 1.2.3 find traits of the graphs when the limit is defined that are not present in the graphs when the limit is not defined. Likewise find traits of the graphs when the limit is not defined that are not present in the graphs when the limit is defined.
\begin{equation*} \lim_{x \to 3\pi} f(x) \end{equation*}
\begin{equation*} \lim_{\gamma \to 1} h(\gamma) \end{equation*}
\begin{equation*} \lim_{\gamma \to 2} h(\gamma) \end{equation*}
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\begin{equation*} \lim_{t \to 1} h(t) \end{equation*}
\begin{equation*} \lim_{t \to 2} h(t) \end{equation*}
\begin{equation*} \lim_{t \to \pi/2} \sigma(t) \end{equation*}
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Figure 1.2.3. Determine which of these limits is defined.
Many of the functions (shapes) we typically deal with are quite nice including that the limit is defined everywhere on them. To construct functions where the limit is not defined in lots of places we need to be creative. These examples, which might be described as pathological, help us more deeply understand the concept.

Checkpoint 1.2.4.

This function, known as the salt and pepper function is an example where the limit is not defined anywhere.
\begin{equation*} sp(x) = \begin{cases} 0 \amp \text{ if } x \text{ is irrational} \\ 1 \amp \text{ if } x \text{ is rational} \end{cases} \end{equation*}
The construction below shows us that between any two points with \(y=1\) is a point with \(y=0\) and vice versa.

(a)

Select two rational numbers \(a\) and \(b_0\) with \(a \lt b_0\text{.}\) Note the function has value 1 at both these points.

(b)

Identify an irrational number \(r_1\) between your two rational numbers (\(r_1 \in (a,b_0)\)). Note the function is 0 at this point (between the two where it is 1).
Hint.
To find an irrational number between yours start with an irrational you know like \(\pi\) or \(e\) or \(\sqrt{5}\) then divide it by something to find one between your pair.

(c)

Identify a new rational number \(b_1\) such that \(a \lt b_1 \lt r_1\) (between your left endpoint and irrational number). Note the function is 1 at this point.

(d)

Identify a new irrational number \(r_2\) between your two rational numbers (\(r_2 \in (a,b_1)\)). Note the function is 0 at this point (between the two where it is 1).

(e)

How many times can this process of finding new rational/irrational numbers closer to \(a\) be repeated?

(f)

As a result can we ever get close enough to \(a\) so that all the points are close to height 1? What does this imply about the limit?

Checkpoint 1.2.5.

For this version of the salt and pepper function the limit is defined at exactly one point \(x=0\text{.}\) Explain why.
\begin{equation*} p_{sp}(x) = \begin{cases} 0 \amp \text{ if } x \text{ is irrational} \\ x^2 \amp \text{ if } x \text{ is rational} \end{cases} \end{equation*}

Checkpoint 1.2.6.

(a)

Create a version of the salt and pepper function such that the limit is defined at exactly two points.

(b)

Create a version of the salt and pepper function such that the limit is defined at every integer but nowhere else.
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