Epsilon delta representation of sum of partial derivaticves

Let $f(x,y)$ be a differentiable function. Can we write $$ \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=\lim_{\epsilon \to 0}\frac{f(x+\epsilon,y+\epsilon)-f(x,y)}{\epsilon} $$

We have $$ \frac{\partial f}{\partial x}=\lim_{\epsilon \to 0}\frac{f(x+\epsilon,y)-f(x,y)}{\epsilon}$$ And $$ \frac{\partial f}{\partial y}=\lim_{\epsilon_1 \to 0}\frac{f(x+\epsilon,y+\epsilon_1)-f(x,y)} {\epsilon_1} $$

How do I combine this to get the above equation. In particular how do I deal with $\epsilon$ and $\epsilon_1$?

Given that $f$ is differentiable, the directional derivative can be obtained by an inner product: $$ \frac{\partial f}{\partial \mathbf{v}} = \nabla f \cdot \mathbf{v} $$ if you take $\mathbf{v}=\mathbf{i}+\mathbf{j}=(1,0)+(0,1)=(1,1)$ you get the result.

In particular \begin{align} &\frac{\partial f}{\partial \mathbf{v}}=\lim_{\epsilon\to0}\frac{f(x+\epsilon,y+\epsilon)-f(x,y)}{\epsilon}\\ &\nabla f \cdot \mathbf{v} = \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \end{align}