Average Power Equation
This formula applies to any (power or energy) signal, not just a periodic signal.

You are correct that for a periodic signal you could change the limits of integration to \$t_0\$ and \$t_0+T\$ (for any value of \$t_0\$ you like), change the time period factor to \$1/T\$, and not have to take a limit.

But not every signal is periodic.


  Can someone explain the time period (1/2t)? Why is it not 1/T (T = period of the time varying signal)?


Actually, it's \$1/(2\tau)\$. This is because the limits of integration are \$-\tau\$ and \$\tau\$, so the total time period being integrated over is \$2\tau\$.

It's not \$T\$ because no period \$T\$ has been defined as part of the problem, and we haven't assumed the signal is periodic.