TT is laughing
Serious calculus question

Perhaps a silly question, but suppose I have a differentiable function f(x,y), is the following statement true?
\lim_{\varepsilon \rightarrow 0} \frac{f(x+\varepsilon,y+\varepsilon)}{\varepsilon}=\lim_{\varepsilon \rightarrow 0} \frac{f(x+\varepsilon,y)}{\varepsilon}+\lim_{\varepsilon \rightarrow 0} \frac{f(x,y+\varepsilon)}{\varepsilon}Ok assuming f(x,y)=0 then you are asking if a directional derivative (45 degree angle) is equal to the sum of the partial derivatives. Why would that always be true? It makes no sense at all.

OP's statement is correct. If the function is differentiable (at (x,y)), then one can replace a directional derivative (LHS) with the product of the gradient and the direction vector (RHS). It is true that the direction vector is often taken to have unit length, while here it is (1,1), so that the LHS is a directional derivative scaled by 2^0.5 but the scaling occurs on the RHS as well, and thus the equality still holds.
This is 100% correct.