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?
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.