"##c\rightarrow\infty##" is for physical intuition (for a layperson or a physics student).
A more mathematical sound approach (which avoids "limiting processes" as much as possible)
is to use a dimensionless parameter that I call
$$E=0,$$ which is one option from ##\{-1,0,1\}##
(in my drafts and posters, I call it ##\epsilon^2##)
which is essentially the sign of the dimensionful quantities with units of a squared-inverse-speed
used by
(An inverse quantity is sometimes used to avoid issues of infinity,
and may be more physical than the historically-defined quantity.
Example: the inverse temperature ##\beta##
thermodynamic beta to handle issues of negative temperature.)From
#42 in Why is Minkowski Spacetime Noneuclidean
one can write
##
\left( \begin{array}{c} t' \\ \frac{x'}{c_{light}} \end{array} \right)
=
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1-E\beta^2}} & \frac{E\beta}{\sqrt{1-E\beta^2}}
\\
\frac{\beta}{\sqrt{1-E\beta^2}} &
\frac{1}{\sqrt{1-E\beta^2}} & \end{array}
\right)\\
\left( \begin{array}{c} t \\ \frac{x}{c_{light}} \end{array} \right)
##
where ##\beta=v/c_{light}## where ##c_{light}=3\times10^8\ \mbox{m/s}## [a fixed quantity, playing the role of a convenient conversion constant].
For ##E=0## (galilean) or ##E=+1## (minkowskian) , one could think of ##E## as if it were $$\left(\frac{c_{light}}{c_{max}}\right)^2,$$
as implemented in code, for example,
https://www.desmos.com/calculator/kv8szi3ic8 .
(As I said in
#58 , this "accounting" approach disentangles
- "c" as a space-time unit conversion constant [which is an issue of history].
- "c" as maximum-signal-speed [which is an issue of physics]
)
So, by primarily using this parameter ##E## or its equivalent [as used above],
we can avoid (or at least minimize) issues of taking limits to infinity
and move on to the other likely-more-interesting mathematical structures of the physics problem.
