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

$$\lim_{\vec x \to \vec x_0} \frac{f(\vec x) - f(\vec x_0) - \nabla f\cdot(\vec x - \vec x_0)}{\|\vec x - \vec x_0\|}=0$$
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.