Answer
Please see the work below.
Work Step by Step
Let us suppose object 1 moves with a velocity $v_{13}$ relative to object 3. If the object 2 moves along the same straight line with a velocity $v_{21}$ relative to object 1, the velocity of object 2 relative to object 3 is $v_{23}$ and
$v_{23}=\frac{v_{21}+v_{13}}{1+\frac{v_{21}v_{13}}{c^2}}$......eq(1)
If the speed of light is infinitely larger, then $c\rightarrow \infty $, which implies that $\frac{1}{c}\rightarrow 0$.
$\implies \frac{1}{c^2}\rightarrow 0$
$\implies \frac{v_{21}v_{13}}{c^2}\rightarrow 0$
$\implies 1+\frac{v_{21}v_{13}}{c^2}\rightarrow 1$
We plug in this value in eq(1) to obtain:
$v_{23}=v_{21}+v_{13}$
Thus, the velocities are added by simple addition in the limit $c\rightarrow \infty$.
