lim [(n+3)/(n+1)]^(n-2)
=lim [1+2/(n+1)]^(n-2)
=lim {[1+2/(n+1)]^[(n+1)/2]}^[(n-2)×2/(n+1)]
=lim e^[2(n-2)/(n+1)]
=e^lim 2(n-2)/(n+1)
=e^lim [2(1-2/n)/(1+1/n)]
=e²