Approximation for high SNR
For coherent BPSK, Taylor series expansion for is as follow,

If X = SNR, then its expansion for [X/(1+X)]^1/2 is,
=  1 - 1/2X + O(1/X^2), now put into your equation of P(e),
You will get as you stated.

Note:


  
    for all approximation you have to expand SNR equation through this formula,
     (aX + b)^n = a^n.X^n - n.a^(n-1).b.X^(n-1) ... n.a.b^(n-1).X + b^n 
  


Using "a" "b" and "n" to stand for any coefficient or exponent we might have