Limits Review: A Rigorous Approach

Limits Review: A Rigorous Approach


Dr. Anton–Calculus Textbook–Quote:
“Example 1 is about as easy a limit proof can get; most limit proofs require a little more algebraic and logical ingenuity. (sic: read basic knowledge should be well understood.) The reader who finds “δ – ε” discussions hard going should not become discouraged; the concepts and techniques are intrinsically difficult. In fact, a precise understanding of limits evaded the finest mathematical minds for centuries.”[3]”[1]


I have hopes to learn some Numerical Methods[2] to allow me to contribute to PLOTS ( Numerical methods are  often used by engineers to mathematically solve problems that do not have “exact” solutions. In truth, they can be used to solve problems that have exact solutions where the solution is quite difficult as well.

Also, the studying of math and engineering help me mentally by providing the psychotherapy methods of mindfulness and compensatory cognitive training for my schizoaffective disorder (bipolar type).

Here, I have provided my first LaTeX created PDF on this topic– LaTeX is great for writing mathematics and is free–that reviews limits and absolute values. I am writing such documents because it is likely that I will lose focus or be hospitalized since that has happened in the past. I can then read a document I made to provide an understanding to move forward. In this case, one of my numerical methods Books started with a vague and abstract definition of the rigorous limits approach[1;2], and I wanted a better understanding[1].

I plan to do similar PDFs for other “review” topics in Section 1.1[2]. By the way, I picked[2] because MIT has a free course that uses the same textbook.

In my reference[1], I also provide a great book that covers the use of Octave and MATLAB when doing numerical methods.

Great Video and Special Note

This MIT video on the Rigorous approach has great information of absolute values, etc. In fact, I suggest the reader pay particular attention, within the video[3], of the power of a simple absolute value property of “Triangle inequality”

|a + b| \leq |a| + |b|

that is discussed during the problem solving near video time of 35:12 to end at 46 minutes.

The professor is being kind. To be good, one must have an amazing memory and an amazing grasp of the fundamentals of mathematics. Really, I believe one should obviously begin this grasp in each math class, but, if one is intelligent enough to grasp future fundamentals ahead of the curve, then he or she can go back and review past sections as they pertain to the current set of problems. I am not that person, but hope to gain enough of a “working” knowledge that I can be competent and useful in the world of PLOTS. With that said, all the video[Embedded; 3] is important and a good test of one’s knowledge on this topic[1].



[1] Harding, Chris. (2015, Dec. 21). Review of Limits. Retrieved (2015, Dec. 21). Available from:; Google URL:

[2] Mathews, John H.; Fink, Kurtis, D. (1999). Numerical Methods Using Matlab. (Third Edition). Prentice Hall

[3] MIT OpenCourseWare. (2011, May 05). Unit I: Lec 5 | MIT Calculus Revisted: Single Variable Calculus, Unit !: Lecture 5: A More Rigorous Approach to Limits. Retrieved (2015, Dec. 28). youtube[online]. Available from: