Alternative Algebra:
Figure I.B.5
GeoGebra Mapping Diagram

Instead of using $x$ and $a$ for the two times used to estimate the instantaneous velocity at time $t=a$, the difference $\Delta t = x-a$ is often represented as $h$ ,  i.e., $h = \Delta t =x-a$.
Using this notation, $x = a+h$ and $\Delta s = s(a+h)-s(a)$. See Figure I.B.5.
As $x \rightarrow a$, $ h \rightarrow 0$ and conversely, as $h\rightarrow 0, x \rightarrow a$ assuming $h \ne 0$.

So the algebra used for estimating $v(a)$ involves using $\frac {\Delta s} {\Delta t} = \frac {s(a+h) - s(a)} h= \frac {(a+h)^2 - a^2} h =\frac {a^2+2ah+h^2 -a^2} h = \frac {2ah +h^2} h =2a +h$.
So as $h \rightarrow 0$ , $\frac {\Delta s} {\Delta t} =2a +h \rightarrow 2a$. Again we see that $v(a)=2a$.