How can the result of that limit be expressed completely?

Question

I want to find the expression of the following limit:

Limit[Power[1 + RealAbs[x]^(3 n), (n)^-1], n -> Infinity]
(* ConditionalExpression[1, (x >= 0[And]log(x)<0)[Or](x<0[And]log(-x)<0)] *)

But the result is not complete. The reference answer is $f(x)=lim _{n rightarrow infty} sqrt[n]{1+|x|^{3 n}}=left{begin{array}{c}
1,|x| leq 1
|x|^{3},|x|>1
end{array}right.$.

What can I do to get the full expression?

calculus and analysis

user64494 · Answer

Let us consider the opposite case by Limit[Power[1 + RealAbs[x]^(3 n), (n)^-1], n -> Infinity, Assumptions -> (x >= 0 && Log[x] >= 0) || (x < 0 && Log[-x] >= 0)] $$text{ConditionalExpression}left[ begin{array}{cc} { & begin{array}{cc} -x^3 & x<0 x^3 & text{True} end{array} end{array} ,x<0lor log (x)>0right] $$ It remains to consider the case RealAbs[x]==1: Limit[Power[1 + RealAbs[x]^(3 n), (n)^-1] /. RealAbs[x] -> 1, n -> Infinity] $1$

How can the result of that limit be expressed completely?

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