The last time I taught the first semester of calculus, I decided to go the route of teaching them limits of functions via characterization by sequences. I found that many students were able to grasp the concepts much better. Since we teach a lower level Real Analysis course, they get the $\varepsilon$-$\delta$ definition there. My issue was that I couldn't find a textbook that used this approach. I had to resort to TeXing some notes up to supplement the class. This became too time consuming. Are there any Calculus texts that define limits using sequences?
$\begingroup$ How did you define limits using sequences? $\endgroup$– Daniel HastMay 10, 2016 at 17:15
2$\begingroup$ Do you want to define convergence of sequences rigorously or just use convergence of sequences as an "intuitively understood concept" without proper proofs? A rigorous approach using sequences seems to essentially replace $\epsilon$-$\delta$ with $\epsilon$-$N$, which may indeed be helpful. $\endgroup$– Joonas IlmavirtaMay 10, 2016 at 17:29
$\begingroup$ @JoonasIlmavirta I would teach them the intuitive definition through examples of sequences with and without limits. But, would probably spend at least half of a class discussing the $\epsilon-N$ definition, so that those continuing on in math will have seen it. $\endgroup$– J126May 10, 2016 at 19:00
4$\begingroup$ @JoeJohnson126: It sounds like you have a definition in mind that you already used. What is this definition? (I can think of several ways to define limits of functions using sequences, which is why I'm asking which approach you're using.) $\endgroup$– Daniel HastMay 10, 2016 at 22:19
1$\begingroup$ This is a little tongue-in-cheek, but you might want to at least look at Calculus Unlimited (full text here: authors.library.caltech.edu/25054/1/CalcUnlimited.pdf). They don't use limits at all (hence the title) but in the final chapter, they relate everything they've done back to the standard definitions for limits, continuity, and differentiation. I don't think you should use the text in your course, but it might help you generate some ideas or give you a different perspective to share with your students. $\endgroup$– Brendan W. SullivanMay 11, 2016 at 0:51
Here is first year calculus text which uses sequences to define continuity of functions: http://www.synechism.org/wp/difference-equations-to-differential-equations/
I found the exposition in this book to be very clear and every step to the concept of the limit is made explicit.