#lim_(x->0)((3+x)/(3-2x))^(1/x)=lim_(x->0)((3-2x+2x+x)/(3-2x))^(1/x)=#
#=lim_(x->0)(1+(3x)/(3-2x))^(1/x)=A#
Let #(3x)/(3-2x)=1/t#, then:
#3tx=3-2x => x(3t+2)=3 => 1/x=(3t+2)/3=t+2/3#
It's obvious that when #x->0# then #t->oo#.
#A=lim_(t->oo)(1+1/t)^(t+2/3)=#
#=lim_(t->oo)(1+1/t)^(2/3) * lim_(t->oo)(1+1/t)^t=#
#=1 * e=e#
Note:
#t->oo => 1/t->0#
#lim_(t->oo)(1+1/t)^t =e#
Note 2:
#lim_(x->0)((3-2x+2x+x)/(3-2x))^(1/x)=#
#lim_(x->0)((3-2x)/(3-2x)+(2x+x)/(3-2x))^(1/x)=#
#lim_(x->0)(1+(3x)/(3-2x))^(1/x)#