What do you mean, with a limit? Like how the derivative is defined for just 1 variable?
This definition actually uses limits, since the limit of e1 (c.q. e2) for Δx (c.q. Δy) going to 0 has to be 0. How does this definition "lack rigor"?
You may generalize this definition to functions .
Then the function f which is defined on a neighborhood of
is differentiable in a if there exists a vector with
Note that this is a multi-dimensional limit and m is exactly the gradient of f in a.
This definition can then also be generalized to functions