Why we should concern with A∞?
Let's suppose a hypothetical irreversible reaction:
$$\ce{B + H -&gt; D + F}$$
where H is a reactant with constant concentration during the reaction and that don't absorb light in the working frequencies. The absorbance of a species $i$ related with the concentration by the Lambert-Beer Law:
$$A_{i}=c_{i}\varepsilon_{i}l$$
where $c_i$ is the concentration of species $i$, $\varepsilon_{i}$ is the molar absorptivity and $l$ is the length of the cuvette.

At an arbitrary time $t$, the total absorbance of the solution is given by:

$$A_{t}=c_{B,t}\varepsilon_{B}l+c_{D,t}\varepsilon_{D}l+c_{F,t}\varepsilon_{F}l
\tag{1}$$

by the stoichiometry of the reaction, we have: $c_{D,t}=c_{F,t}=c_{B,0}-c_{B,t}$; where $c_{B,0}$ is the initial concentration of the reactant B, so replacing this in Eq. 1 and defining a auxiliary constant $\varepsilon_{i}l=k_{i}$, we have:

$$A_{t}=c_{B,t}k_{B}+k_{D}(c_{B,0}-c_{B,t})+k_{F}(c_{B,0}-c_{B,t})$$
$$A_{t}=c_{B,t}(k_{B}-k_{D}-k_{F})+c_{B,0}(k_{D}+k_{F})\tag{2}$$

At a time $t\rightarrow \infty$  (a time long enough that the reaction goes to completion), $c_{B,\infty}=0$ and $c_{D,\infty}=c_{F,\infty}=c_{B,0}$, so the absorbance is given by:

$$A_{\infty}=c_{D,\infty}k_{D}+c_{F,\infty}k_{F}$$
$$\boxed{A_{\infty}=c_{B,0}(k_{D}+k_{F})} \tag{3}$$

so,replacing Eq.3 in Eq.2 its follows that:

$$A_{t}=c_{B,t}(k_{B}-k_{D}-k_{F})+A_{\infty}$$
$$\boxed{A_{t}-A_{\infty}=c_{B,t}(k_{B}-k_{D}-k_{F})}$$

In conclusion: $A_{\infty}$ is important because is directly proportional to the initial concentration of reactant B. When you subtract $A_{\infty}$ from the absorbance at an arbitrary time $t$ $(A_{\infty})$ it allows you to know (indirectly) the concentration of the reactant at the aforementioned time $t$.