Relativistic Limit

Question

Excuse me.

I have a problem about double limit, namely, Relativistic Limit.

Let $c$ is positive real constant. Calculate \begin{equation} \lim_{v\to c}\lim_{V\to c}\frac{v - V}{1 - vV/c^2}. \end{equation} I have tried in many times, but I fail.

Thank you for the answer.

Answer 1

We have

$$ \lim_{v\rightarrow c }\> \lim_{V\rightarrow c }\frac{v-V}{1-vV/c^2} = \lim_{v\rightarrow c }\frac{v-c}{1-v/c}= \lim_{v\rightarrow c }-c =-c. $$