I will assume the limit to be: #lim_(a->0) (a+x)^(3/a) #
First, consider that #a-> 0^+# then #3/a# increases beyond all bounds.
Thus #lim_(a->0^+) (a+x)^(3/a) -># x raised to positive infinity.
#:. if absx> 1 -> lim_(a->0^+) (a+x)^(3/a) = +oo#
and # if absx< 1 -> lim_(a->0^+) (a+x)^(3/a) = 0#
also, #if x={0,1} ->lim_(a->0^+) (a+x)^(3/a)# is undefined.
We can also analyse the limit as #a->0^-#
In this case #lim_(a->0^-) (a+x)^(3/a) -># x raised to negative infinity.
#:. if absx> 1 -> lim_(a->0^-) (a+x)^(3/a) = 0#
and # if absx< 1 -> lim_(a->0^-) (a+x)^(3/a) = +oo#
again, #if x={0,1} ->lim_(a->0^-) (a+x)^(3/a)# is undefined.
So we can see that the limit yields different results as #a# approaches #0# from above or below for all #x# where the limit is defined.
Hence, the limit does not exist.
