# Calculus Without Limits

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About *Calculus Without Limits:*

Excerpts from book:

I love calculus! This love affair has been going on since the winter of 1966 and, perhaps a little bit before. Indeed, I remember purchasing my first calculus textbook (by Fobes and Smyth) in December of 1965 and subsequently pouring through its pages, pondering the meaning of the new and mysterious symbols before me. Soon afterwards, I would be forever hooked and yoked as a student, teaching assistant, teacher, and lifelong admirer.

Over the years, my rose-colored perspective has changed. I have discovered like many other instructors that most students don’t share an “aficionado’s” enthusiasm for calculus (as we do). The reasons are many, ranging from attitude to aptitude, where a history of substandard “classroom-demonstrated” mathematical aptitude can lead to poor attitude. The tragedy is that with some students the aptitude is really there, but it has been covered over with an attitude years in the making that says, “I just can’t do mathematics.” These students are the target audience for this book.

I once heard it said by a non-engineer that an engineer is a person who gets excited about boring things. Not true! As an engineer and educator myself, I can tell you that an engineer is a person who gets excited about very exciting things—good things of themselves that permeate every nook and cranny of our modern American culture. The problem as the warden in the Paul Newman movie Cool Hand Luke so eloquently stated, “is a failure to communicate.” The volume in your hands, Calculus Without Limits, is a modern attempt to do just that—communicate! Via a moderate sum of pages, my hope is that the basic ideas and techniques of calculus will get firmly transferred to a new generation, ideas and techniques many have called the greatest achievement of Western science.

The way this book differs from an ordinary “encyclopedicstyle” textbook is twofold. One, it is much shorter since we cover only those ideas that are central to an understanding of the calculus of a real-valued function of a single real-variable. The shortness is also due to a lack of hundreds upon hundreds of skillbuilding exercises—very necessary if one wants to become totally competent in a new area of learning. However, a minimal set of exercises (about 200 in all) is provided to insure that the reader can verify understanding through doing. Two, as stated by the title, this is a calculus book that minimizes its logical dependence on the limit concept (Again, Chapter 4.). From my own teaching experience and from reading book reviews on web sites, the limit concept seems to be the major stumbling block preventing a mastery of engineering-level calculus. The sad thing is that it doesn’t need to be this way since calculus thrived quite well without limits for about 150 years after its inception; relying instead on the differential approach of Newton and Leibniz.

Differentials—little things that make big ideas possible—are the primary means by which calculus is developed in a book whose title is Calculus Without Limits. The subtitle —Almost refers to the fact that the book is not entirely without limits. Section 4.3 provides an intuitive and modern explanation of the limit concept. From that starting point, limits are used thereafter in a handful (quite literally) of key arguments throughout the book .

Now for the bad news! One, Calculus Without Limits is a primer. This means that we are driving through the key ideas with very few embellishments or side trips. Many of these embellishments and side trips are absolutely necessary if one wants a full understanding of all the technical power available in the discipline called calculus. To achieve full mastery, nothing takes the place of all those hours of hard work put into a standard calculus sequence as offered through a local college or university. This book should be viewed only as an aid to full mastery—a starter kit if you will. Two, Calculus Without Limits is not for dummies, morons, lazy bones, or anyone of the sort. Calculus Without Limits is for those persons who want to learn a new discipline and are willing to take the time and effort to do so, provided the discipline is presented in such a matter as to make in-depth understanding happen. If you don’t want to meet Calculus Without Limits halfway—providing your own intellectual work to understand what is already written on each page—then my suggestion is to leave it on the book-seller’s shelf and save yourself some money.

I love calculus! This love affair has been going on since the winter of 1966 and, perhaps a little bit before. Indeed, I remember purchasing my first calculus textbook (by Fobes and Smyth) in December of 1965 and subsequently pouring through its pages, pondering the meaning of the new and mysterious symbols before me. Soon afterwards, I would be forever hooked and yoked as a student, teaching assistant, teacher, and lifelong admirer.

Over the years, my rose-colored perspective has changed. I have discovered like many other instructors that most students don’t share an “aficionado’s” enthusiasm for calculus (as we do). The reasons are many, ranging from attitude to aptitude, where a history of substandard “classroom-demonstrated” mathematical aptitude can lead to poor attitude. The tragedy is that with some students the aptitude is really there, but it has been covered over with an attitude years in the making that says, “I just can’t do mathematics.” These students are the target audience for this book.

I once heard it said by a non-engineer that an engineer is a person who gets excited about boring things. Not true! As an engineer and educator myself, I can tell you that an engineer is a person who gets excited about very exciting things—good things of themselves that permeate every nook and cranny of our modern American culture. The problem as the warden in the Paul Newman movie Cool Hand Luke so eloquently stated, “is a failure to communicate.” The volume in your hands, Calculus Without Limits, is a modern attempt to do just that—communicate! Via a moderate sum of pages, my hope is that the basic ideas and techniques of calculus will get firmly transferred to a new generation, ideas and techniques many have called the greatest achievement of Western science.

The way this book differs from an ordinary “encyclopedicstyle” textbook is twofold. One, it is much shorter since we cover only those ideas that are central to an understanding of the calculus of a real-valued function of a single real-variable. The shortness is also due to a lack of hundreds upon hundreds of skillbuilding exercises—very necessary if one wants to become totally competent in a new area of learning. However, a minimal set of exercises (about 200 in all) is provided to insure that the reader can verify understanding through doing. Two, as stated by the title, this is a calculus book that minimizes its logical dependence on the limit concept (Again, Chapter 4.). From my own teaching experience and from reading book reviews on web sites, the limit concept seems to be the major stumbling block preventing a mastery of engineering-level calculus. The sad thing is that it doesn’t need to be this way since calculus thrived quite well without limits for about 150 years after its inception; relying instead on the differential approach of Newton and Leibniz.

Differentials—little things that make big ideas possible—are the primary means by which calculus is developed in a book whose title is Calculus Without Limits. The subtitle —Almost refers to the fact that the book is not entirely without limits. Section 4.3 provides an intuitive and modern explanation of the limit concept. From that starting point, limits are used thereafter in a handful (quite literally) of key arguments throughout the book .

Now for the bad news! One, Calculus Without Limits is a primer. This means that we are driving through the key ideas with very few embellishments or side trips. Many of these embellishments and side trips are absolutely necessary if one wants a full understanding of all the technical power available in the discipline called calculus. To achieve full mastery, nothing takes the place of all those hours of hard work put into a standard calculus sequence as offered through a local college or university. This book should be viewed only as an aid to full mastery—a starter kit if you will. Two, Calculus Without Limits is not for dummies, morons, lazy bones, or anyone of the sort. Calculus Without Limits is for those persons who want to learn a new discipline and are willing to take the time and effort to do so, provided the discipline is presented in such a matter as to make in-depth understanding happen. If you don’t want to meet Calculus Without Limits halfway—providing your own intellectual work to understand what is already written on each page—then my suggestion is to leave it on the book-seller’s shelf and save yourself some money.