# $\lim_{t\to +\infty}\frac{d}{ds}f(t,s)=g'(s)?$

Supose that $$f:(0,\infty) \times (0,\infty) \to \mathbb{R}$$ is a $$C^2((0,\infty) \times (0,\infty))$$ function, and $$\lim_{t\to +\infty}f(t,s)=g(s)$$, can we conclude that $$\lim_{t\to +\infty}\frac{d}{ds}f(t,s)=g'(s)?$$ I am trying use the Mean Value Theorem but i cant any argument.

Let $$f:(0,\infty) \times (0,\infty) \to \mathbb{R}$$ be defined as $$f(t,s)=\frac{\sin(t^2s)}{t}$$. Then $$\lim_{t\to \infty}f(t,s)=0$$ but $$\lim_{t\to \infty}f_s(t,s)= \lim_{t\to \infty} t \cos(t^2s)$$ does not exist.
• and supposing that $\lim_{t\to \infty} f_s(t,s)$ exist? Oct 14 at 0:03