One possibility is the way I just happen to be presenting the material. I thought I'd take a moment and record a couple of examples here.

- To introduce limits, I started by asking how we might graph $f(x) = \frac{sin(x)}{x}$. We had previously talked about the graph of $f(x) = \frac{1}{x}$, and what happens at $x = 0$. We said that while you cannot plug in 0, you can ask what happens near 0. So we tried that again. What happens to $f(x)$ as $x$ gets closer and closer to 0. We made a table, everyone agreed that the $y$-values were getting closer to 1. Only then did I start using the language "limit." I gave them the notation as a way to quickly write down what we just did. This seemed much more natural than giving them a definition of a limit (out of thin air) and then showing them a bunch of examples.
- For left/right-hand limits, I only introduced them after we discovered the problem in finding limits if the $y$-values approach different values from each side. I gave them a graph and asked them to find limits in a variety of cases. We all agreed that at the jump discontinuity, the limit did not exist. But of course, we can be more descriptive than this. Coming from one side, the limit does exist. Then I introduced that notation.
- For continuity, I drew two graphs, one continuous, and one with a jump discontinuity. I asked the class what was different about the graphs. We agreed that the continuous one was continuous and the other not. I asked what other ways a graph might not be continuous. We came up with a graph with a removable discontinuity and an infinite discontinuity. Now, how might we say something about these in terms of limits?
- Continuing with continuity, to introduce the difference between continuous at a point vs interval, I covered up the discontinuity and asked if the rest of the graph was continuous. We agreed it was, so decided that we needed to express continuity at specific points. This led us to the limit definition of continuity. This was so much more natural than giving the definition, and then figuring out what it means.

I don't know if this is helping the students (we have not had any exams yet) but it definitely feels better to me. I plan to make an effort to continue using this "technique" for the rest of the class and see how it goes.