So far no one's mentioned (or did I miss it?) that if you make students compute $$ \lim_{w\to 5} \frac{w^6 - 5^6}{w - 5}, $$ then some of them will use L'Hopital's rule to do that, if you don't tell them not to.

Here's an example of something I have students do with the limit definition of the derivative: http://wnk.hamline.edu/~mjhardy/1170/notes/quiz.10.19.pdf

They find all sorts of creative ways of getting things wrong when doing this. Here's another: http://wnk.hamline.edu/~mjhardy/1170/homework/13th.pdf

I think after they've done several like this, they actually do learn what this is for, and that it's not being used as a way to avoid quick and efficient ways of computing derivatives.

But I have them thinking about instantaneous rates of change without using limits on the first day of the course: http://wnk.hamline.edu/~mjhardy/1170/handouts/September.8.pdf