A clarification regarding partial derivatives Let us suppose the $i^{th}$ partial derivative of $f:\Bbb{R}^n\to \Bbb{R}$ exists at $P$; i.e. if $P=(x_1,x_2,\dots,x^n)$, $$\frac{f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)}{\Delta x_n}=f'_n (P)$$
My book says this implies that $$f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)=f'_n(P)\Delta x_n + \epsilon_n \Delta x_n$$
such that $\lim\limits_{\Delta x_n\to 0} \epsilon_n=0$. 
I don't understand where $\epsilon_n$ comes into the picture. Why can't we just have $f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)=f'_n(P)\Delta x_n$, considering we're anyway using $\Delta x_n$ as a real number rather than an operator. 
Justification for asking on overflow- I'm doing research on multi-variable calculus..?
Thanks!

Compare the definition of the derivative to your ratio when $f:\mathbb R\to\mathbb R$, $x\mapsto x^2$.
