13. Introduction to Limits (Switching Dialects from our Usual Infinitesimal Chatter)

Now that we've reached (almost) the end of our journey through differential calculus, we'll give our infinitesimal workhorses a rest. While they are enjoying their well-deserved oats, we'll see what these lithe and lively limits can do. We'll redo some work using limits instead of infinitesimals to see how the limits will handle the task. In particular, we'll see what calculus looks like if we think of derivatives 𝐧𝐨𝐭 as ratios of infinitesimals, but rather as limits of ratios of 𝘳𝘦𝘢𝘭 numbers. As we'll see, everything turns out exactly the same, only expressed in stodgier language.

This is the thirteenth in a sequence of supplementary videos for a calculus class that I teach based on Chapters 1, 2, 3, and π of my book 𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠.

𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠,
𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎,
𝑃𝑟𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 𝑀𝑎𝑑𝑒 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡,
𝐿𝑜𝑏𝑎𝑐ℎ𝑒𝑣𝑠𝑘𝑖 𝐼𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑.

The first three are available for sale as paperbacks at Amazon, and as pdfs at Lulu. (The Lobachevski book is available at Amazon and the American Mathematical Society)

0:00 Intro
0:37 A Rough Definition
1:24 Five Examples from a Graph
11:23 Seven Examples from Formulas
18:02 The Limit Definition of a Derivative
25:45 Computing a Derivative From the Limit Definition...
32:50 ...Compared to the Infinitesimal Approach.
34:47 Second Example of a Derivative via the Limit Definition

[gustavodelgadillo]

This type of videos for real analysis would be a gem in the dessert, superb exaplanation as usual.