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Mathematica does not evaluate any of these limits. It seems that the answers should be $1/x$ and $1/y$, respectively. 

Limit[(x^k + y^k)^(-1/k), k -> ∞, Assumptions -> x > 0 && y > 0 && x > y]

Limit[(x^k + y^k)^(-1/k), k -> ∞, Assumptions -> x > 0 && y > 0 && x < y]

They seem to be fairly simple limits. Am I missing something here?

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    $\begingroup$ My guess is that Limit treats the expression as a complex-valued function, even though the assumptions imply otherwise. In particular, transformations such as this are not used: PowerExpand[Factor /@ PowerExpand[(x^k + y^k)^(-1/k) /. y -> (u x)]] /. u -> y/x. $\endgroup$
    – Michael E2
    May 9, 2017 at 22:28
  • $\begingroup$ Limit often fails in cases that seem simple. Try Series instead. $\endgroup$
    – bbgodfrey
    May 9, 2017 at 23:29
  • $\begingroup$ In this case Series performs no better. $\endgroup$
    – bbgodfrey
    May 10, 2017 at 1:27

1 Answer 1

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The limits evaluate for x > 1 && y > 1 for which the larger term dominates

Limit[(x^k + y^k)^(-1/k), k -> Infinity, 
 Assumptions -> x > 1 && y > 1 && x > y]

(*  1/x  *)

Limit[(x^k + y^k)^(-1/k), k -> \[Infinity], 
 Assumptions -> x > 1 && y > 1 && x < y]

(*  1/y  *)
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