Let $f(x)$ be a continuously differentiable function such that $f(x) \to 0$ as $x \to \infty$. It is known that this does not necessarily imply that $df(x)/dx \to 0$ as $x \to \infty$.
Consider now $g(y,x): D_y \times D_x \to \mathbb{R}$, $D_y \subseteq \mathbb{R}$, $D_x \subseteq \mathbb{R}$. Assume that $g(y,x)$ is continuously differentiable, and $g(y,x) \to 0$ as $x\to \infty$ $~~\forall y \in D_y$. What can be said about $$\lim_{x\to \infty} \frac{\partial g(y,x)}{\partial y}.$$ Under what conditions will it converge to zero? Note that the partial derivative is taken w.r.t. $y$.