I am having the function $$ f(x,y,z) = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$ and I need to find the directional derivative at point $(1,-1,1)$ for every $\vec{v}$ $\in{\mathbb{R}^3}$ by using the definition of the partial derivative. and also I need to the determine what is the max and min that the $\partial f(1,-1,1)$ at any $\vec{v}=(v_1,v_2,v_3)$ can have.
So my problem is first to calculate the limit by definition, I am getting an expression like $$ \lim_{t\to 0} \frac{\frac{1}{1+tv_1} + \frac{1}{tv_2-1} + \frac{1}{1+tv_3} -1}{t\sqrt{v_1^2 +v_2^2 + v_3^2}}$$
I have no idea how to go on from here, and also, how I should understand the second part of the question, about the minimum and maximum of the directional derivative at the point $(1,-1,1)$
thank you kindly.