Interchange of limit and derivative of bounded convex function on half space

Question

Let $u\in C^2((0, \infty)^2; \mathbb{R})$ be a convex and bounded and let \begin{align} f(y)\colon= \lim_{x\to \infty}u(x, y), ~~~y >0. \end{align} Then, can I guarantee that $f$ is continuously twice differentiable and \begin{align} f'(y) =& \lim_{x\to \infty}u_y(x, y),\\ f''(y) =& \lim_{x\to \infty}u_{yy}(x, y)? \end{align}