compound interest; taking the limit

I cannot follow all the steps in taking the limit of the compound
interest formula below. I lose it when derive the variable m and
substitute it in the equation. If Icould get some help with that it
would be great. I understand everything before and after the m
substitution, except how they derived it.
Take the limit o f A=P(1+ r/n)^nt
first move the limit inside: lim_n^infinity[p(1+r/n)^nt] =
p{lim_n^infinity[(1+r/n)^n]}^t
Do the above to associate the limit with independant variable n.
In the next step the above is = p{lim_m^infinity[(1+1/m)^m]}^rt
They explain in the book the above equation is found by letting m=n/r
which leads to r=nm to establish what they call a towing relation****p
where n goe to infinity and m goes to infinity. In the end they arrive
at the limit of compound interest formule being p{e}^rt. I just really
don't understand how m comes to be.