Counterexample of $\lim_{x\rightarrow \infty}f(x,t(x))\neq 1$ where $\lim_{x\rightarrow \infty }t(x)=2$.

Question

Given $\lim_{x\rightarrow \infty} f(x,t)=1$ for any fixed t in $(1,3)$, in general $\lim_{x\rightarrow \infty} f(x,t(x))\neq 1$ where $\lim_{x\rightarrow\infty}t(x)=2$ and $t(x)\in (1,3)$ unless we have uniform convergence. However, I cannot find a good counterexample. Any suggestion is highly appreciated. Thank you very much.

Answer 1

What about this one?$$f(x,t)=\begin{cases}1+{{\tan{\pi\over 4}t}\over x}&,\quad t\ne 2\\1&,\quad t=2\end{cases}$$and $t(x)=2-e^{-x}$.