Why are epsilon-delta proofs difficult?

The epsilon-delta definition of a limit is often a student's first exposure to universal and existential quantifiers in a formal setting. It's important to understand each one and how "for all, there exists" differs from "there exists, for all". It was not immediately apparent to me and others in my class that these two statements are wildly different.

Getting the quantifiers straight is also important for helping students understand which variable to "fix" and which to "choose". When the definition says "there exists a $\delta$", students can confuse that as something that is given and that they somehow need to prove that this mysterious $\delta$ works for all $\epsilon$.

I think a personal problem I had with epsilontics when first learning it that the proofs tend to be not very indicative of how one can arrive at them. One chooses values according to some inequality and verifies that it works out. The process of actually arriving at the inequalities involved is hidden in such proofs, but essential to creating them.

So I think it is particularly hard to learn epsilontics by simply working through a lot of epsilontics. Later one develps some intuitions and heuristics for getting to these inequaities, but its hard to explicate them. Also, this is one of the first occurances of proofs requiring a significant amount of creativity, they are not purely mechanical.

Among the difficulties, I feel that students have trouble with the logic. When trying to construct a $\delta$ for a given $\epsilon$, it is often necessary to work backwards. You then notice that all your inequalities were if and only if. So, you can work them in the reverse, and logically valid, direction. Students tend not to understand what it means to assume what you are trying to prove.

Then, when presented with an example that cannot be worked out in the reverse direction, they try to do so. When you tell them that will not work, they often have trouble trying to figure out how to even start the problem. They lack the tricks to find the appropriate $\delta$.

I am a student, and I always find epsilon-delta proofs hard. The reason is the notation, and the number of things to keep in your head at the same time. It is always something like:

For any epsilon, there exists a delta, such that it holds that ...

Once you get to really understand the underlying mechanics, the epsilon/delta proofs are not that hard - they are just really scary looking!