I'm also interested in defective definitions that may not have been proposed in writing, but are the sort that someone might have come up with en route to the correct definition. E.g., "We say that $\lim_{x \rightarrow a} f(x) = L$ if for every number $\epsilon > 0$ there is a corresponding number $\delta$ such that if $0 < |x-a| < \delta$ and $f(x)$ is defined, then $|f(x)-L| < \epsilon$." (Under this definition, the limit of sqrt(x) as x approaches -1 is anything you please!) As I explained to my students on the first day of class: In a non-honors course, one treats calculus as a vehicle for getting from point A to point B. In an honors course, one wants the students to poke around under the hood and find out what the different parts do by removing them from the vehicle and then driving the vehicle off a cliff to see what happens. (Nearly all of the students in the class are declared engineering majors, and I think they find this motivation for rigor more appealing than fussy appeals to intellectual hygiene.) Jim Propp