Wednesday, 10 September 2014

multivariable calculus - Intuition behind differentiability

I've just started studying multi-variable calculus and I found myself mumbling a lot over the definition of differentiability. Here's the one I'm using. A function if differentiable if
lim
Now, I recall that the gradient gives the direction of maximum increase of the function, and I understand that the term with the gradient gives, in fact, the component of the gradient in the direction of the vector \vec x - \vec x_0. What I do not understand is why, rearranging the limit, the limit for \vec x \to \vec x_0 of \lim_{\vec x \to \vec x_0} \frac{f(\vec x) - f(\vec x_0)}{\|\vec x - \vec x_0\|} has to be precisely
\frac{\nabla f\cdot(\vec x - \vec x_0)}{\|\vec x - \vec x_0\|}Thank you in advance for your time.

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