## Definition of a Derivative/Limit

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

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:

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