Well, the definition of derivative is probably one of the best application of the notion of limit, from a didactical point of view. If you define the derivative as a limit process then students who understand it will not miss the geometric flavour: the slope of the tangent line is the limit as $h\rightarrow 0$ of the slope of the line through $(x, f(x))$ and $(x+h, f(x+h))$. I think this is beautiful and relatively simple, once you get the students to think about it for a minute. Plus, it answers the question "When do we agree that the graph of $f$ admit the existence of a tangent line at $(x, f(x))$"?

Of course one has to keep in mind that for most students the useful thing to learn is how to compute practically a derivative without using the definition but rather applying a collection of rules. Nevertheless I think it is important to give them an idea of where all these rules come from. Think about those students who want to get a a math major? No?

In Italy in the so called "scientific high school", the schools that provide you with the widest and most basic education (you learn a bit of everything) with a focus in math, physics, chemistry perhaps, ecc.. we are thought the limit using the $\epsilon-\delta$ definition, and the derivative from its definition. This is to say that I think it is possible to have students learn this theoretical aspects of calculus, if high school kids do.

Well, the definition of derivative is probably one of the best application of the notion of limit, from a didactical point of view. If you define the derivative as a limit process then students who understand it will not miss the geometric flavour: the slope of the tangent line is the limit as $h\rightarrow 0$ of the slope of the line through $(x, f(x))$ and $(x+h, f(x+h))$. I think this is beautiful and relatively simple, once you get the students to think about it for a minute. Plus, it answers the question "When do we agree that the graph of $f$ admit the existence of a tangent line at $(x, f(x))$"?

Of course one has to keep in mind that for most students the useful thing to learn is how to compute practically a derivative without using the definition but rather applying a collection of rules. Nevertheless I think it is important to give them an idea of where all these rules come from. Think about those students who want to get a a math major? No?

In Italy in the so called "scientific high school", the schools that provide you with the widest and most basic education (you learn a bit of everything) with a focus in math, physics, chemistry perhaps, ecc.. we are taught the limit using the $\epsilon-\delta$ definition, and the derivative from its definition. This is to say that I think it is possible to have students learn this theoretical aspects of calculus, if high school kids do.