This is a "reverse" question of finding the asymptote of a function

Recently, I am interested in doing some sort of modelling which involve equations of the form


where $f(t)$ is required to satisfy the following properties

$$1^*:\left\{\begin{matrix}f(0)=0 \\ \lim_{t\rightarrow \infty}f(t)=1 \\\text{f(t) is not of the form g(t)+1 where}\lim_{t\rightarrow \infty}g(t)=0 \end{matrix}\right. $$

That is, $@(t)$ is modelling the change of some quantity (e.g. population) that starts at some constant value and slowly decay to zero as time progresses. I am interested in finding $f$ that satisfy $1^*$ so that I can experiment on various decay behaviour.

I know the graphical solution to $1^*$ is basically any functions that passes through the origin and has an asymptote y=1 in the posive t direction and does not involve some other function shifted up by 1, for example the following (x is t but google does not allow plotting using t as a variable) enter image description here

However, it may not be easy to obtain a graphical solution, or even if at least one graphical solution can be found, (I have no idea how to) obtain the mathematical expression for them for more general problems such as

$$2^*:\left\{\begin{matrix}\lim_{t\rightarrow\infty}h(t)=c ,\text{where c}\in \mathbb{R} \text{and is given} \\ \lim_{t\rightarrow\infty}[h(t)-k(t)]=0 ,\text{where k(t) is given}\\\lim_{t\rightarrow a}h(t)-\lim_{t\rightarrow b}h(t)=4,\text{where a,b}\in \mathbb{R} \text{and are given}\\\ln(\lim_{t\rightarrow a}h(t))=8, \text{where c}\in \mathbb{R} \text{and is given} \end{matrix}\right. $$


Q1. What are the common, algorithmic or analytical approaches in solving systems of limit equations?

Q2. For the particular case where some finite number of asymptote functions were given, how to recover the original function and under what condition is the solution unique?

Q3. How to obtain a mathematical expression from the graphical solutions to these problems?


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