When you are a student, rather than a researcher, formal definitions must come first, and intuition can be developed later. Otherwise, you are just allowing yourself to say nonsense.
I worry about the fact that you're being downvoted, because what you're saying is utterly crucial and important. I fear those who ignore it will waste a lot of time with really misguided ways of doing mathematics. The analogy approach works for programming, but it's not applicable to seriously studying mathematics.
I wouldn’t dismiss analogy so easily. George Polya opened my eyes to the power of analogy in math. I can’t recommend his “Mathematics and Plausible Reasoning” enough.
Polya's text was targeted towards researchers and people who already understood the basics pretty well. But I don't think the analogy approach is valid (or at least, not recommended) in the context of this thread, where the parent seemed to assume that the audience wasn't even particularly comfortable with limits. It can be a dangerous and seductively-easy path to go down, is all I'm saying. You're not going to understand mathematics without having a firm firm grasp of the rigorous definitions and by doing mountains of exercises (where it will often be quickly apparent that using an analogy is not sufficient).
I wouldn't worry too much about it. Math students are not the primary target audience of this website. And, if the planets align and someone here actually wants to study math, they will either quickly discover the limitations of an “analogy-first” approach, or quickly fail and give up.
Analogies are always leaky abstractions and it's easy to cement an incorrect notion of the subject matter that doesn't account for subtle behavior and edge cases. I do think explaining by analogy is useful as long as it comes with clear caveats, but as an example, if there were good analogies for quantum uncertainty and wave particle duality, or general relativity, people wouldn't find those things eternally perplexing. YouTube has thousands of hours of videos trying to explain GR with elevators in space and trains moving at the speed of light, but if you can't do the math you'll never really understand it. Some things just can't be broken down into concepts that a three dimensional monkey brain can easily understand.
