Predicting time taken to reach equilibrium

Consider the following reaction, $\ce{A <=>[k\_f][k\_b] B}$, $k\_f$ and $k\_b$ are the rate constants of forward and backward reactions respectively.
Let the initial concentration of $\ce{A}$ be $a \pu{M}$ and the final equilibrium concentrations would be $(a-x\_e)\pu{M}$ and $x\_e\pu{M}$ respectively, where $$\frac{k\_f}{k\_b} = \frac{x\_e}{a-x\_e}$$i.e,$$x\_e = \frac{k\_fa}{k\_f+k\_b}$$
Let $x\pu{M}$ $\ce{A}$ be consumed at any time, $t$, then the *net* rate of formation of $\ce{B}$ is
$$\frac{dx}{dt} = k\_f(a-x) - k\_b(x)$$
Solving for $x$, with limits from $x=0$ to $x=x$ and $t=0$ to $t=t$
$$\frac{1}{k\_f + k\_b} ln(\frac{k\_fa}{k\_fa - (k\_f+k\_b)x})=t$$
Substituting $x=x\_e$, $t \to \infty$, but in many books, it is shown that reactions reach equilibrium after a certian time like the following


[![From NCERT, CHEMISTRY PART-1](https://i.stack.imgur.com/sRoeU.png)](https://i.stack.imgur.com/sRoeU.png)


Image taken from [NCERT Text book for Chemistry, Class XI, Part-1](https://ncert.nic.in/ncerts/l/kech107.pdf).


[![From Chemistry Libre Texts](https://i.stack.imgur.com/57CQU.png)](https://i.stack.imgur.com/57CQU.png)


Image taken from [Chemistry Libre Texts](https://chem.libretexts.org/Courses/University_of_Kentucky/UK%3A_CHE_103_-_Chemistry_for_Allied_Health_(Soult)/Chapters/Chapter_8%3A_Properties_of_Solutions/8.2%3A_Chemical_Equilibrium).


A google search result:
[![Attainment of Chemical Equilibrium](https://i.stack.imgur.com/aI9UK.png)](https://i.stack.imgur.com/aI9UK.png)


All the above images show that equilibrium state is attained at a *finite* time but mathematically it takes $\infty$ time to reach equilibrium.


Is there any flaw in the equation or in the graphs?




The flaw is in the graph if you take its labels in their literal sense.


[![enter image description here](https://i.stack.imgur.com/5gcvJ.jpg)](https://i.stack.imgur.com/5gcvJ.jpg)


So technically, the label "at equilibrium" is incorrect for the left panel. You can either weaken the label language (center) or change the graph (right) to have a watertight graph and labels.


Source of original graph: <https://chem.libretexts.org/Courses/University_of_Kentucky/UK%3A_CHE_103_-_Chemistry_for_Allied_Health_(Soult)/Chapters/Chapter_8%3A_Properties_of_Solutions/8.2%3A_Chemical_Equilibrium>


 2