Skip to main content
\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\fe}[2]{#1\mathopen{}\left(#2\right)\mathclose{}} \newcommand{\cinterval}[2]{\left[#1,#2\right]} \newcommand{\ointerval}[2]{\left(#1,#2\right)} \newcommand{\cointerval}[2]{\left[\left.#1,#2\right)\right.} \newcommand{\ocinterval}[2]{\left(\left.#1,#2\right]\right.} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\fd}[1]{#1'} \newcommand{\sd}[1]{#1''} \newcommand{\td}[1]{#1'''} \newcommand{\lz}[2]{\frac{d#1}{d#2}} \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} \newcommand{\lzo}[1]{\frac{d}{d#1}} \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\sech}{\operatorname{sech}} \newcommand{\csch}{\operatorname{csch}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Activity2.1Limits

While working Exercise Group 1.3.1.18–1.3.1.22 from Activity 1.3 you completed Table 1.3.3. In the context of that problem the difference quotient being evaluated returned the average rate of change in the volume of fluid remaining in a vat between times \(t=4\) and \(t=4+h\). As the elapsed time closes in on \(0\) this average rate of change converges to \(-6\). From that we deduce that the rate of change in the volume \(4\) minutes into the draining process must have been \(6\) galmin.

Please note that we could not deduce the rate of change \(4\) minutes into the process by replacing \(h\) with \(0\); in fact, there are at least two things preventing us from doing so. From a strictly mathematical perspective, we cannot replace \(h\) with \(0\) because that would lead to division by zero in the difference quotient. From a more physical perspective, replacing \(h\) with \(0\) would in essence stop the clock. If time is frozen, so is the amount of fluid in the vat and the entire concept of “rate of change” becomes moot.

It turns out that it is frequently more useful (not to mention interesting) to explore the trend in a function as the input variable approaches a number rather than the actual value of the function at that number. Mathematically we describe these trends using limits.

If we denote \(\fe{f}{h}=\frac{\fe{V}{4+h}-\fe{V}{4}}{h}\) to be the difference quotient in Table 1.3.3, then we could describe the trend evidenced in the table by saying “the limit of \(\fe{f}{h}\) as \(h\) approaches zero is \(-6\).” Please note that as \(h\) changes value, the value of \(\fe{f}{h}\) changes, not the value of the limit. The limit value is a fixed number to which the value of \(\fe{f}{h}\) converges. Symbolically we write \(\lim\limits_{h\to0}\fe{f}{h}=-6\).

Most of the time the value of a function at the number \(a\) and the limit of the function as \(x\) approaches \(a\) are in fact the same number. When this occurs we say that the function is continuous at \(a\). We will explore the concept of continuity more in depth at the end of the this lab, once we have a handle on the idea of limits. To help you better understand the concept of limit we need to have you confront situations where the function value and limit value are not equal to one another. Graphs can be useful for helping distinguish function values from limit values, so that is the perspective you are going to use in the first couple of problems in this lab.

Subsection2.1.1Exercises

1

Several function values and limit values for the function in Figure 2.1.1 are given below. You and your group mates should take turns reading the equations aloud. Make sure that you read the symbols correctly, that's part of what you are learning! Also, discuss why the values are what they are and make sure that you get help from your instructor to clear up any confusion.

\(\fe{f}{-2}=6\), but \(\lim\limits_{x\to-2}\fe{f}{x}=3\).

\(\fe{f}{-4}\) is undefined, but \(\lim\limits_{x\to-4}\fe{f}{x}=2\).

\(\fe{f}{1}=-1\), but \(\lim\limits_{x\to1}\fe{f}{x}\) does not exist.

\(\underbrace{\lim_{x\to1^{-}}\fe{f}{x}}_{\begin{array}{c}\text{the limit of }\fe{f}{x}\\\text{as }x\text{ approaches }1\\\text{from the left}\end{array}}=-3\), but \(\underbrace{\lim_{x\to1^{+}}\fe{f}{x}}_{\begin{array}{c}\text{the limit of }\fe{f}{x}\\\text{as }x\text{ approaches }1\\\text{from the right}\end{array}}=-1\).

Figure2.1.1\(y=\fe{f}{x}\)

Copy each of the following expressions onto your paper and either state the value or state that the value is undefined or doesn't exist. Make sure that when discussing the values you use proper terminology. All expressions are in reference to the function \(g\) shown in Figure 2.1.2.

Figure2.1.2\(y=\fe{g}{t}\)
2

\(\fe{g}{5}\)

3

\(\lim\limits_{t\to5}\fe{g}{t}\)

4

\(\fe{g}{3}\)

5

\(\lim\limits_{t\to3^{-}}\fe{g}{t}\)

6

\(\lim\limits_{t\to3^{+}}\fe{g}{t}\)

7

\(\lim\limits_{t\to3}\fe{g}{t}\)

8

\(\fe{g}{2}\)

9

\(\lim\limits_{t\to2}\fe{g}{t}\)

10

\(\fe{g}{-2}\)

11

\(\lim\limits_{t\to-2^{-}}\fe{g}{t}\)

12

\(\lim\limits_{t\to-2^{+}}\fe{g}{t}\)

13

\(\lim\limits_{t\to-2}\fe{g}{t}\)

Values of the function \(f\), where \(\fe{f}{x}=\frac{3x^2-16x+5}{2x^2-13x+15}\), are shown in Table 2.1.3, and values of the function \(p\), where \(\fe{p}{t}=\sqrt{t-12}\), are shown in Table 2.1.4. These questions are in reference to these functions.

\(x\) \(\fe{f}{x}\)
\(4.99\) \(2.0014\)
\(4.999\) \(2.00014\)
\(4.9999\) \(2.000014\)
\(5.0001\) \(1.999986\)
\(5.001\) \(1.99986\)
\(5.01\) \(1.9986\)
\(t\) \(\fe{p}{t}\)
\(20.9\) \(2.98\ldots\)
\(20.99\) \(2.998\ldots\)
\(20.999\) \(2.9998\ldots\)
\(21.001\) \(3.0002\ldots\)
\(21.01\) \(3.002\ldots\)
\(21.1\) \(3.02\ldots\)
Table2.1.3\(\fe{f}{x}=\frac{3x^2-16x+5}{2x^2-13x+15}\)
Table2.1.4\(\fe{p}{t}=\sqrt{t-12}\)
14

What is the value of \(\fe{f}{5}\)?

15

What is the value of \(\lim\limits_{x\to5}\dfrac{3x^2-16x+5}{2x^2-13x+15}\)?

16

What is the value of \(\fe{p}{21}\)?

17

What is the value of \(\lim\limits_{t\to21}\sqrt{t-12}\)?

Create tables similar to Tables 2.1.3 and 2.1.4 from which you can deduce each of the following limit values. Make sure that you include table numbers, table captions, and meaningful column headings. Make sure that your input values follow patterns similar to those used in Tables 2.1.3 and 2.1.4. Make sure that you round your output values in such a way that a clear and compelling pattern in the output is clearly demonstrated by your stated values. Make sure that you state the limit value!

18

\(\lim\limits_{t\to6}\dfrac{t^2-10t+24}{t-6}\)

19

\(\lim\limits_{x\to-1^{+}}\dfrac{\sin(x+1)}{3x+3}\)

20

\(\lim\limits_{h\to0^{-}}\dfrac{h}{4-\sqrt{16-h}}\)