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I am confused about things about "iterated limit" and "double limit"

I am doing something about complex variables, and learning to derive the Cauchy-Riemann Equation.

The concept of "iterated limit" to me is that, supossing 2 independt variables x&y, it is taking the limit either x first then y or y first then x. Geometrically, it is representing two paths to approach the right destination, right?

But I can't really visualise how "double limit" really work.

What is a double limit? I am told that it is "x and y go togather in any manner"

What does it mean?

I can think of some cases that x and y will approach the destination togather.

Say, the path is y=mx as x->0. In this case, x and y will approach to 0 togather. So, is this a double limit?

If this is, there is also a problem.

There is a statement that "the limit, representing the derivative of a complex function, must exist as a double limit for delta z= delta x+i(delta y) approaching zero."

I don't see why the term "double limit" is used here.

To me, the complex derivative exists when all double limits and all iterated limits gives the same value so that the limit can be said of being independent of any paths. Am I correct?