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Section 6.2 Limits of the form zero over zero: Which section are you referring to?

This section starts with the sentences

"In the last section, we were interested in the limits we could compute using continuity and the limit laws. What about limits that cannot be directly computed using these methods?"

But in my comments on Section 6.1, I point out that the limits posed in the questions in Section 6.1 cannot be computed using the limit laws. In Section 5.3, students did compute limits using limit laws.

There is some confusion here because of your ambiguous use of the word "Section". Throughout the book, clarity would be improved if you used the words "Chapter" and "Section" for the Appropriate things. For example, Chapter 5 ought to have five sections:

Chapter 5 Limit Laws
Section 5.0 Learning Outcomes (the thing that you have numbered as 5)
Section 5.1 Equal or Not
Section 5.2 Continuity
Section 5.3 The Limit Laws
Section 5.4 Continuity of Piecewise Functions

Section 6.3 Limits of the form nonzero over zero: Small comment about wording

I really like that you devoted a section of the book to this topic. I have a couple of comments. The first one is minor, about wording.

In your definition of the infinite limit, you say

"...the limit of f(x) approaches infinity as x goes to a."

To me, this does not sound right. The limit does not approach infinity. The limit is infinity.

I would say that the words

"the limit of f(x), as x approaches a, is infinity"

are defined to mean

"the values of f(x) go to infinity as x approaches a"

The reason I like this better is that the words

"the limit of f(x), as x approaches a, is infinity"

are an exact translation into prose of the notation lim_{x \rightarrow a} f(x) = \infty

Section 2.2 Example about why sqrt(x^2) = abs(x)

The presentation of this function is in symbols, and the function is not given the name f(x). You used the name f(x) to denote the square root function in the previous example. So when you say that x is a positive number, and you ask the reader to find f(-x), the answer really should be DNE. Perhaps use the name g(x) for the function sqrt(x^2), and ask the reader to find g(-x). And change the name of the graph to y = g(x).

In Section 2.4, you pose a yes/no question "Is f(x) invertible at x=0.5?" I have a couple of problems with this.

(1) You have not said what it means for a function to be invertible. This is easy enough to remedy.

(2) You have not said what it means for a function to be invertible at a particular x-value. I don't think this is a commonly-used expression. Reading your question, I wondered if perhaps you were asking if there is a y-value such that f^-1(y) = 0.5. The answer to this question would be "no". But the book says that "yes" is the correct answer.

So I wonder if what you are asking the reader is, "Does f^-1(0.5) exist?". This wording of the question works for mathematicians who know that, whether or not f is invertible, the symbol f^-1(y) is well-defined and denotes the preimage of y. But it does not work in the current context, because f^-1 has so far only been used to denote the inverse function, and for the current example, f^-1 does not exist. So maybe simply ask more clearly, "Is there an x value such that f(x) = 0.5?". Or "Suppose y=0.5. Is there an x-value such that f(x)=y?"

See ximeraTutorial/titlePage.tex

Section 2.1 Example of ln(x+1)

Your parser recognizes the input "log_e(x+1)" and typesets it correctly, but it is rejected as an incorrect answer.

But the input "log(x+1)" is accepted as correct.

I would think that the first answer should certainly be considered as correct.

If anything, the second answer should be considered as problematic, given that in some realms, log(x) is used to denote log base 10.

See understandingFunctionsB/digInForEachInputExactlyOneOutput.tex

In Section 2.2 For each input, exactly one output, you pose the following question.

Question: Which of the following are functions?

You use the word "mapping" in the question, but you use the word "relating" in your discussion of the correctness or incorrectness of the various answers. The same word should be used in both places. (And again, I don't think the terminology of relations is necessary or helpful for the target audience.)

Section 2.3 The Explanation about the driving distance example

The sentence "...we first relate how far one can drive..." is confusing and therefore not helpful. In terms of the sequence of events in the composition f(g(m)), one first determines g(m), and then determines f(g(m)).

Section 6.1 Could it be anything?: Puzzling Abstract

Abstract says "Two young mathematicians investigate the arithmetic of large and small numbers." But they don't seem to be discussing the arithmetic of large and small numbers, at least not in those terms. There are some questions posed in Section 5.1 Equal or Not about dividing large and small numbers.

Section 5.3 The Limit Laws: graph of f(x) = sqrt(ln(x))

Be clearer about what the reader should be observing in the graph of f(x) = sqrt(ln(x))

Perhaps observe that the graph appears to be heading for the location (x,y) = (e,1), and highlight this with some annotations on the graph.

And I'm confused here as I was earlier: why do you use an embedded, active graph for your illustration? Why not just a static graph?

Section 4.1 Stars and Functions: motivating limits

The dialogue about stars is actually misleading and therefore not helpful. It relates the idea of looking to the side of a dim star in order to see the star. Limits are sort of about looking to the side of x=c, but it is not for the purpose of determining the y value at x=c. Rather, it is for identifying the trend in the y-values when x is not equal to c.

When the dialogue is about the function f(x), it is incorrect as worded: What is happening with the function f(x) at x=2 is simply that f(2) DNE. Furthermore, I think it does not do a very good job of motivating the upcoming material on limits.

My approach to motivating the coming material on limits would be something like the following:

I think it would be good to start with a graph of the rational function f(x) that you have introduced, f(x) = (x-1)(x-2)/(x-2).

Observe that there is no y-value at x=2, because f(2) DNE.

But observe that there is a "trend" in the graph near x=2: The graph appears to be heading for the location (x,y) = (2,1)

Reiterate that there is no point on the graph at the location (x,y) = (2,1), because we have already determined that f(2) DNE.

Suggest that it is natural to wonder if there is some mathematical way of describing the fact that the graph appears to be heading for the location (x,y) = (2,1)

Introduce the limit as the precise mathematical language that will be used to articulate the idea that the graph appears to be heading for the location (x,y) = (2,1).

Reiterate that the function value f(a) has to do with the behavior precisely at x = a, while the limit of a function is about the trend in the y-values when x is near a but not equal to a.

Section 2.3 Examples about and x^2 and \sqrt{x}

You changed the names of the functions in the second example. That's too bad. Would be better to stick with f(x)=x^2 and g(x) = \sqrt{x}. Then you can observe that f \circ g \neq g \circ f.

See understandingFunctionsB/digInInversesOfFunctions.tex

Section 2.4 Inverse Relations

You state the inverse relations nicely as

f(g(b) = b for all b in B
g(f(a) = a for all a in A

but then a few lines later, you convert everything to the variable x:

f(f^-1(x)) = x
f^-1(f(x)) = x

I think it would be clearer if you retained the variable names that correspond to the sets where those variables live, and continue to state the sets.

f(f^-1(b)) = b for all b in B
f^-1(f(a)) = a for all a in A

or

f(f^-1(y)) = y for all y in B
f^-1(f(x)) = x for all x in A

Section 5.2 Continuity: Your sentence motivating the idea of continuity is problematic in a number of ways.

To start with, you say

"...plugging the value into the function."

This ought to say more clearly

"...x-value...."

rather than just

"...value...."

But more importantly, at this point in the text, students have not computed any limits by using the formula for a function. They have only found limits by looking at graphs. So your motivation for the concept of continuity really ought to refer to graphs. Here is the way that I motivate it in my course. (My introduction has elements of the "lift the pencil" idea that you refer to later, when discussing continuity on an interval.)

Introduction to Limits that I use in my course

Observe that we have seen some behavior of graphs that is obviously "weird"

Hole in graph at x = a, with missing dot
Hole in graph at x = a, with dot in wrong place
Jump in graph at x = a.

A unifying informal description of all of these could be that one must lift the pencil to draw the graph in the vicinity of x = a. (One might think that one does not have to lift the pencil if there is hole in the graph, but rather can just go around the hole like a roundabout in a road and then proceed to the rest of the graph. But in fact, the open circle at the location (x,y) = (a,L) is meant to make it clear that there is no dot at that location. So strictly speaking, one would draw the graph by approaching x=a from the left, lifting the pencil over the location (x,y) = (a,L), and then putting it down to the right of that location and proceeding with the rest of the graph.)

On the other hand, speaking informally, much of the graph can be drawn without having to lift the pencil from the page.

Is there some more precise, more concise, way of articulating the idea of being able to draw the graph in the vicinity of x = a without having to lift the pencil?

There is, using the terminology of limits and function values. The concept is given a name: "continuity".

Definition: To say that a function is continuous at x = a means that the following three things are true
(1) f(a) is defined. That is, there is a point on the graph of f(x) at the location (x,y) = (a,f(a))
(2) The limit of f(x), as x approaches a, exists. That is, the graph appears to be heading for some location (x,y) = (a,L)
(3) The limit of f(x), as x approaches a, equals f(a). That is, the location (x,y) = (a,L) that the graph appears to be heading for is the same location (x,y) = (a,f(a)) where there is a point on the graph.

Informally, one could say that a function is continuous at x = a if the graph of f(x) can be drawn near x = a without lifting the pencil.
******************** End of introduction to Limits that I use in my course *****************

Section 6.2 Limits of the form zero over zero: Please explain more clearly.

I my calculus courses, students are forever confused about the issue of when one can and cannot cancel factors in an expression like f(x) = (x-1)(x-2)/(x-2)

Near the top of the page, in the first example, you say note that if we assume x is not equal to 2, then we can cancel terms. I think this needs much more and much clearer explanation. Why can we "assume" that x is not equal to 2?

Here's how I do it:

In my courses, I stress to the students that the most important concept of the first month of the course is the idea that when computing f(2), one cannot cancel factors and so f(2) DNE, while when computing the limit, as x approaches 2, of f(x), one does cancel.

I compute f(1), f(2), f(3), and show how numbers in the numerator do cancel when computing f(1) and f(3). But I point out that f(2) leads to 0/0, which is undefined. One cannot cancel 0/0.

Then I discuss that when when the limit, as x approaches 2, of f(x), one is supposed to consider the values of f(x) when x is close to 2 BUT NOT equal to 2. Since x is not equal to 2, we know that x-2 is not equal to 0, so we can cancel (x-2)/(x-2). We are definitely NOT cancelling 0/0 here. We are cancelling terms whose value we don't know, except that we do know the terms are NOT zero. I tell them that I want them to be able to explain that cancellation step that clearly, because it is the most important concept of the first month of the class.

Follow the link below to a set of lecture notes for one of my class meetings. Pages 9 - 17 discuss an example similar to the one in your book. (Of course you'll write whatever you want--it's your book--but I do think that a book's discussion of this concept should be as thorough as my lecture notes.)

https://people.ohio.edu/barsamia/2019-20.1.1350/lecture.notes/Day.02.pdf

Section 2.4 Your Theorem about a function being one-to-one at x=a

You have not defined what it means for a function to be one-to-one at a particular x-value. I have never seen that concept defined, and I don't think such a concept is necessary.

Perhaps say "A function is one-to-one if and only no horizontal line intersects the curve at more than one point on the domain."

Section 3.3 Polynomial Functions: Define Whole Number or just say "integer \geq 0". Many students don't remember what "whole number" means, and if they do a web search, the top result they will find is wrong.

See reviewOfFamousFunctions/digInPolynomialFunctions.tex

See understandingFunctionsB/digInFinancialMathematics.tex

Section 2.6 Definitions of Demand function and Supply function are awkwardly worded.

Section 4.2 What is a limit? Clarify the definition of limit

In your definition of limit, I would suggest adding a phrase saying that "a" and "L" are real numbers.

And since your definition of limit is an "intuitive" one, I would suggest adding an additional sentence describing the corresponding graphical behavior. Say that the words

"the limit of f(x) as x approaches a is L"

correspond to the graphical behavior

"the graph appears to be heading for the location (x,y) = (a,L)"

and maybe even point out that this does not say anything at all about what happens at x = a.

Section 2.4 Your question about a function being one-to-one on an interval

You have not defined what it means for a function to be one-to-one on an interval.

Section 6.2 Limits of the form zero over zero: Add parentheses to an expression

In the problem where they have to factor the denominator of (x-1)/(x^2 + 2x - 3), the answer x+3 is accepted as correct. But the answer needs to be (x+3), in parentheses. Either put parentheses around the box where students are to type their answer, or require that there be parentheses in the students' answer.

I think it is not a good idea to allow sin theta without the parentheses. In general, I think it is a good idea to encourage students to put parentheses around the arguments of functions.

Section 2.6 Financial Mathematics: types of Compound Interest

Add Definition of Simple Interest A = P(1+rt)

Say "Periodically-Compunded Interest" for the function A = P(1+r/n)^(nt), on order to distinguish it from "Continuously-Compounded Interest" that (presumably) will make an appearance at some point in the book.

(Glancing at the table of contents, I'm unable to determine where Continuously Compounded Interest might show up in the book. But I'm assuming that it will show up at some point.)

Section 4.2 What is a limit? mistake in the graph of f(x) = x^2/4x

Hole in the graph of f(x) = x^2/4x should be at (x,y) = (0,0), not at (x,y) = (0,1)

Section 3.4 Exponential and logarithmetic Functions

Domain & Range of Exponential & Logarithmic function

I think it would be helpful to give both the domain and range of the exponential functions in their definitions.

Section 6.1 Could it be anything?: How are students supposed to do the problems?

Three problems are posed about function values at x = 0 and limits at x = 0 for functions involving x/x. The limit laws of Section 5.3 do not apply. Discussion of Limits of the form zero over zero does not happen until the upcoming Section 6.2. So how do you envision students answering these questions in Section 6.1? Do you want them to make graphs? Instructions are not clear.

Section 6.3 Limits of the form nonzero over zero:

I really like that you devoted a section of the book to this topic. I have a couple of comments. The first one is minor, about wording.

In your definition of the infinite limit, you say

"...the limit of f(x) approaches infinity as x goes to a."

To me, this does not sound right. The limit does not approach infinity. The limit is infinity.

I would say that the words

"...the limit of f(x), as x approaches a, is infinity"

are defined to mean

"...the values of f(x) go to infinity as x approaches a"

The reason I like this better is that the words

"...the limit of f(x), as x approaches a, is infinity"

are an exact translation into prose of the notation lim_{x \rightarrow a} f(x) = \infty

See indeterminateForms/digInLimitsOfTheFormNonZeroOverZero.tex

See understandingFunctionsB/breakGroundB.tex

Section 2.1 Same or Different?

You write

"...The domain of a function is part of the ‘‘data’’ of the function. A function is not a rule for transforming the input to the output, but rather the relationship between a specified collection of inputs (the domain) and possible outputs (the range)...."

This is odd wording. For mathematicians who know about relations, we know what these sentences are about. But for students who have not studied relations, the sentences are too vague to be helpful. (See note on another issue created in Section 2.2)

Section 3.4 Exponential and logarithmetic Functions: typo in section title

See reviewOfFamousFunctions/digInExponentialAndLogarithmeticFunctions.tex

Section 2.6 Equilibrium Point

Before talking about the Equilibrium Point, you should discuss the Demand Curve and the Supply Curve some more. Observe that the Demand Curve will be decreasing and the Supply Curve will be increasing. (Discuss why.) Include crude sketches as part of this observation. Then observe that those curves will intersect at one point.

Section 6.2 Limits of the form zero over zero: Consider limits of difference quotients

The last two examples, involving the limit of (1/(x+1) - 3/(x+5))/(x-1) and the limit of (sqrt(x+5)-2)/(x+1) are difficult examples, and they will not show up again in the book. But they are comparable in form and to the limits that one would compute when computing the f'(x) or f'(c) for f(x) = 1/(x+5) or f(x) = sqrt(x+5). I suggest that in Section 6.2, you use limits that will also appear in future sections when computing derivatives. That way, students will get to see the same hard limit more than once.

(In fact, I would suggest breaking Section 6.2 into two sections, with the second section devoted to the limits of difference quotients.)

Section 4.2 What is a limit? Suggested addition to the discussion "Limits might not exist"

Regarding your discussion of the limits of the Greatest Integer function.

Earlier, I suggested that you add to your intuitive definition of limit the remark that the words

"the limit of f(x) as x approaches a is L"

correspond to the graphical behavior

"the graph appears to be heading for the location (x,y) = (a,L)"

If you do make that addition, then your discussion of the non-existence of the limits for the greatest integer function could end with a remark about that idea. Currently, that discussion ends with the sentence

"We cannot find a single number that f(x) approaches as x approaches 2, and so the limit does not exist[s]."

To this, you could add

"Put another way, as x approaches 2, there is not a single location (x,y) = (2,L) that the graph appears to be heading for. So the limit does not exist."

Section 2.2 For each input, exactly one output

The Definition of function is too vague and not complete. Your definition of function should include the terms "domain" and "range", especially in light of your remark in section 2.1 that the "...domain of a function is part of the ‘‘data’’ of the function....".

You use the terminology of relations. If you are going to use that terminology, it should be introduced and it should be used more precisely. In my Business Calculus course, I use an informal definition of function that does not use the terminology of relations, but that is much more precise than your definition. (Personally, I feel that for the target audience, there is no need to use the word "relation". The students in a course like this have not studied relations, and they are not going to study relations in the future.)

And I think that here, at the place where "function" is defined, is the place to introduce the idea of equality of functions. That is, discuss here that a "function" consists of a domain, a codomain, and a relation from the domain to the codomain. To say that two functions f,g are the same means that those three items are the same for f,g. And remark that the same relation can sometimes be presented in ways that look different but that actually describe the same relation. (But again, I would rather not use the terminology of relations.) This will make your later example about f(x) = (x-2)(x-1)/(x-2) and g(x) = x-1 clearer.

Section 5.3 The Limit Laws: Composition Limit Law

The equation x = the limit as x approaches a of g(x) is bad.

Perhaps write the composition law more clearly as follows:

If L = the limit as x approaches a, of g(x), and if f(x) is continuous at x = L, then the limit, as x approaches a, of f(g(x)) is f(the limit, as x approaches a, of g(x)).

That is, the limit, as x approaches a, of f(g(x)) is f(L)

Section 3.4 Exponential and logarithmetic Functions

Description of the base b

I think that symbolic math should the same grammar and syntax as prose. Your definition of an exponential function has the phrase

"...where b \neq 1 is a positive real number."

This reads "...where b is not equal to one is a positive real number."

Would be clearer as "...where b is a positive real number not equal to 1."

Same comment about the definition of logarithmic function.

Section 5 Limit Laws: Learning outcomes are not written in a parallel way.

"Famous functions are continuous on their domains" is not a learning outcome.

Section 5.2 Continuity: "x is continuous at a point a"

Throughout this section, you say a few times

"x is continuous at a point a"

or

"...the points x where a function is not continuous..."

Call me cranky, but "a" is not a point. It is an x-value.

You should say more clearly

"x is continuous at an x-value x = a...."

or

"...the x-values where a function is not continuous...."

Section 2.6 Financial Mathematics Add Definition of Simple Interest A = P(1+rt)

See understandingFunctionsB/digInFinancialMathematics.tex

Section 2.2 Theorem about the vertical line test

You have not defined what it means for a curve to represent a function at a particular x-value. I have never seen that concept defined, and I don't think such a concept is necessary.

Perhaps say "A curve represents a function with domain D if and only if every vertical line of the form x = a, where a \in D, intersects the curve at exactly one point. "

Section 5.4 Continuity of Piecewise Functions

First piecewise-defined function: the domain partitions should just be "if x < 0", and "if x \geq 0"

And I'm confused here as I was earlier: why do you use an embedded, active graph for your illustration? Why not just a static graph?

Section 4.2 What is a limit? System won't accept correct answer for the first question (e)

The answer to question (e) is DNE. The system will not accept this answer.

Section 6.3 Limits of the form nonzero over zero: Enlarging the definition of limit

My second comment about this section is more substantive. I think it is very important to point out that in this section, we are essentially enlarging the definition of limit.

In Section 4.2 What is a Limit, the value of a limit had to be a number. For example, if we were to compute the limit, as x approaches 3 from the left, of f(x) = (x-7)/(x-3), the answer would be that the limit does not exist because the section 4.2 limit laws do not allow us to find the limits of the form nonzero over zero.

But now, in Section 6.3, we say that that same limit does exist: the limit is infinity.

But there is more subtlety here that should be discussed. Using the Section 4.2 limit laws, if we consider the two-sided limit, that is, if we were to compute the limit, as x approaches 3, of f(x) = (x-7)/(x-3), the answer would again be that the limit does not exist because the section 4.2 limit laws do not allow us to find the limits of the form nonzero over zero.

And using Section 6.3 techniques, if we were to compute the limit, as x approaches 3, of f(x) = (x-7)/(x-3), the answer would also be that the limit does not exist. But it does not exist for a different reason. The limit does not exist because the left and right limits don't match.

(Note that nowhere in Section 6.3 do you discuss the alternate formulation of the definition of two-sided limit, in terms of equality of the two one-sided limits.)

(I accidentally closed this issue earlier.)

Your parser recognizes the input "log_e(x+1)" and typesets it correctly, but it is rejected as an incorrect answer.

And the input "log(x+1)" is accepted as correct.

I would think that the first answer should certainly be considered as correct.

If anything, the second answer should be considered as problematic, given that in some realms, log(x) is used to denote log base 10.

Section 6.3 Limits of the form nonzero over zero: Enlarging the definition of limit

My second comment about this section is more substantive. I think it is very important to point out that in this section, we are essentially enlarging the definition of limit.

In Section 4.2 What is a Limit, the value of a limit had to be a number. For example, if we were to compute the limit, as x approaches 3 from the left, of f(x) = (x-7)/(x-3), the answer would be that the limit does not exist because the section 4.2 limit laws do not allow us to find the limits of the form nonzero over zero.

But now, in Section 6.3, we say that that same limit does exist: the limit is infinity.

But there is more subtlety here that should be discussed. Using the Section 4.2 limit laws, if we consider the two-sided limit, that is, if we were to compute the limit, as x approaches 3, of f(x) = (x-7)/(x-3), the answer would again be that the limit does not exist because the section 4.2 limit laws do not allow us to find the limits of the form nonzero over zero.

And using Section 6.3 techniques, if we were to compute the limit, as x approaches 3, of f(x) = (x-7)/(x-3), the answer would also be that the limit does not exist. But the limit does not exist for a different reason. Using Section 6.3 techniques, the limit does not exist because the left and right limits don't match.

(Note that nowhere in Section 6.3 do you discuss the alternate formulation of the definition of two-sided limit, in terms of equality of the two one-sided limits. I think that should be discussed.)

Section 5.3 The Limit Laws: graph of f(x) = 5x^2 + 3x - 2

Be clearer about what the reader should be observing in the graph of f(x) = 5x^2 + 3x - 2

Perhaps observe that the graph appears to be heading for the location (x,y) = (1,6)

And remark that there is a point on the graph at (x,y) = (1,6), but that that is irrelevant. The fact that the limit turns out to be 6 simply conveys the fact that the graph appears to be heading for the location (x,y) = (1,6).

And I'm confused: why do you use an embedded, active graph for your illustration? Why not just a static graph?

Section 4.2 What is a limit? Change the order of the questions

Farther down the page, you ask questions (a) - (l) about a graph.

Your question (j) ought to be put after question (e).

Section 2.4 Your illustration of y = x^2 and y = \sqrt{x} is confusing. You call the blue graph f and the red graph f^-1, but the blue graph is not invertible.

Three possible solutions:

• Only graph f on the restricted domain [0, \infty )
• Graph f on the whole real line, but make the left branch dotted blue instead of solid blue.
• Define a new function g(x) = x^2 on the restricted domain [0, \infty ), and graph g and g^-1

Section 2.6 Defined terms are not clearly presented

Should be clearer that the terms such as Demand, Price, Supply, Cost, Average Cost and Profit are defined terms. Present them as definitions, with some sort of visual highlighting.

Section 2.4 Your "Warning" about the square-root function

It should say that \sqrt{x} is defined to be the "non-negative" (rather than "positive") square root.

And the phrase "Thinking of the square-root as the inverse of the squaring function..." is not helpful and is technically wrong, since the squaring function does not have an inverse. Perhaps word things a little differently, and use a slightly different example.

Maybe say that

y = \sqrt{x} means two things:

y^2 = x
y \geq 0

So for example, the equation y^2 = 9 has two solutions, y = 3 and y = -3, while the equation y = \sqrt{9} has only one solution, y = 3.

You write

"...The domain of a function is part of the ‘‘data’’ of the function. A function is not a rule for transforming the input to the output, but rather the relationship between a specified collection of inputs (the domain) and possible outputs (the range)...."

This is odd wording. For mathematicians who know about relations, we know what these sentences are about. But for students who have not studied relations, the sentences are too vague to be helpful. (See note on another issue created in Section 2.2)

Section 4.2 What is a limit? typo

About midway down the page, the discussion of the Greatest Integer function ends with the sentence

"We cannot find a single number that f(x) approaches as x approaches 2, and so the limit does not exists."

The last word should be "exist".