Definition of a Derivative/Limit

cse63146

I'm in a multi-variable calculus course, and we need to know definitions for the test.

Is this a correct definition of a derivative:

Let [tex]f:R^n \rightarrow R^p[/tex] f is diffrentiable at x iff there exists a matrix (p x n) Df(x) such that [tex]lim (h\rightarrow0)\frac{f(x - h) -f(x) -Df(x)h}{||h||} = 0[/tex]

My professor gave us quite a few definitions for derivative, some involve gradients and epsilon. Just wondering if the above also works.

and is this a correct definition of a limit:

[tex]lim (x \rightarrow x_0) F(x) = A [/tex] there exists [tex]\epsilon > 0[/tex] such that [tex]\delta>0 \Rightarrow [/tex] [tex]0<||x - x_0|| <\delta \Rightarrow ||f(x) - A|| < \epsilon[/tex]