I was wondering if it was possible to find/approximate an equation from just a single known point and the known limit.
For example,
$$f(0) = 0.8$$
$$\lim_{x \to \infty} \hspace{.1cm}f(x) = 1.8$$
I'm not sure what the $x$ will be at any given time. All I know is that when $x = 0, y = .8$ and, as $x$ approaches infinity, the $y$ gets closer to $1.8$.
With such a small amount of information we can't really determine a sufficiently small set of solutions, there are undoubtedly infinitely many functions which satisfy your two criteria, among whom are complicated looking ones such as, $$f(x):= \begin{cases} 0.8, & x=0\\ 0.8+\dfrac{x^4+10^{10^{10}}x^3-x^2+ex+\pi^\pi}{x^4+42824}, & x\neq 0 \end{cases},$$ or simpler ones, such as $f:x\mapsto -1000^{-2x}+1.8$.