A (good) teacher looks at a problem from as many aspects as possible and works with students to understand them. A (good) teacher collects multiple ways to understand a concept, so if one doesn't sink in, the next may. It's not an inborn skill; it takes practice.
The best way I've ever seen limits explained, incidentally, is taking a student 10 feet away from a wall. Ask the student to go half way to the wall. Then again. Then again. Eventually, the student is taking centimeter steps. Ask the student if he's at the wall. The student says yes or no, but usually, there is an "aha!" moment, the idea being that you can figure out where "there" is without ever technically being "there."
Understanding the basics of the derivative was the first place I banged my head, but that was because I didn't adequately understand the secant. Once I understood that, the derivative fell into place. (In fact, calculus was where I realized how much I didn't understand trigonometry.)