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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?

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1 Answer 1

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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)]

$$\text{ConditionalExpression}\left[ \begin{array}{cc} \{ & \begin{array}{cc} -x^3 & x<0 \\ x^3 & \text{True} \\ \end{array} \\ \end{array} ,x<0\lor \log (x)>0\right] $$ It remains to consider the case RealAbs[x]==1:

Limit[Power[1 + RealAbs[x]^(3 n), (n)^-1] /. RealAbs[x] -> 1, n -> Infinity]

$1$

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