Evaluating a seemingly simple limit, using continuity of the partial derivative I am stuck with a seemingly easy problem. I have a function $f(x,y)$ which has continuous first derivatives. Now, I want to show that: $$ \lim_{h \to 0}\dfrac{f(a+hu,b+hv)-f(a,b+hv)}{hu} =f_x(a,b)$$
This is what I have thought: Since the derivatives are continuous, we know that, for each $\epsilon > 0$, there is a $\delta > 0$ such that it is:
$$0 < ((x-a)^2 + (y-b)^2)^{1/2} < \delta \implies |f_x(x,y) - f_x(a,b) | = \left|\lim_{h \to 0}\dfrac{f(x+h,y)-f(x,y)}{h} - \lim_{h \to 0}\dfrac{f(a+h,b)-f(a,b)}{h} \right| < \epsilon$$
My plan was to relate the first expression to this definition of the continuity of partial derivative. But the $hv$ which is added to $b$ spoils this idea. I think I am missing something very obvious here, but annoyingly I can't figure what. How should I proceed?
 A: You need a little care with the value of $u$. You need $u \neq 0$ for the quotient to be defined. I will avoid this by multiplying through by the fixed
quantity $u$.
Using the mean value theorem we have
$f(a+hu, b+hv) -f(a, b+hv) = f_x'(a+t_h hu, b+hv)hu$ for some $t_h \in (0,1)$.
Let $\epsilon >0$, and choose $\delta>0$ such that
$|f_x'(c,d)-f_x'(a,b) | < \epsilon$ for all $\|(c,d)-(a,b)\| < \delta$.
Now suppose $|h| \|(u,v)\| < \delta$ and let $t_h$ be the value above, then
\begin{eqnarray}
|{f(a+hu, b+hv) -f(a, b+hv) \over h} - f_x'(a,b) u| &=& |f_x'(a+t_h hu, b+hv)u - f_x'(a,b) u| \\
&=& |f_x'(a+t_h hu, b+hv) - f_x'(a,b)| | u| \\
&\le& \epsilon |u|
\end{eqnarray}
It follows that
$\lim_{h \to 0} {f(a+hu, b+hv) -f(a, b+hv) \over h} = f_x'(a,b) u$.
A: Observe that by MVT: $f(a+hu,b+hv) - f(a+hu,b) = hv\cdot \dfrac{\partial f}{\partial y}(a+hu,c)$, and $f(a,b+hv) - f(a,b) = hv\cdot \dfrac{\partial f}{\partial y}(a,d)$. Thus we can write:
$\dfrac{f(a+hu,b+hv) - f(a,b+hv)}{hu} = \dfrac{f(a+hu,b) + hv\cdot \dfrac{\partial f}{\partial y}(a+hu,c) - f(a,b) - hv\cdot \dfrac{\partial f}{\partial y}(a,d)}{hu} = \dfrac{f(a+hu,b) - f(a,b)}{hu} + \dfrac{v}{u}\cdot \left(\dfrac{\partial f}{\partial y}(a+hu,c) - \dfrac{\partial f}{\partial y}(a,d)\right) \to f_x(a,b) + 0 = f_x(a,b)$ since when $u \neq 0, hu \to 0 \to h \to 0 \to c,d \to b$
