Suppose $\displaystyle A \subset R^m $. Then $\displaystyle f: A \rightarrow R^n $ is differentiable at a point $\displaystyle \vec{x} \in A $ if there is an n by m matrix $\displaystyle B $ such that $\displaystyle \lim_{h \rightarrow 0} \frac{ ||f(\vec{x} + \vec{h}) - f(\vec{x}) - B \vec{h}||}{||\vec{h}||} $.

My question is, does it matter HOW $\displaystyle \vec{h} $ goes to 0? For example, consider the function $\displaystyle f(x,y) = xy $. Say I wanted to calculate its derivative.

When taking the limit of the derivative difference quotient, is it permissible to let the entries of the matrix $\displaystyle \vec{h} \in R^2$ approach 0 in any fashion I want?

I am asking this because I am just learning multi-variable calculus, and when calculating the derivatives I've always let $\displaystyle \vec{h} $ = (h h) and was wondering if I am allowed to do this.