4
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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 '17 at 22:28
  • $\begingroup$ Limit often fails in cases that seem simple. Try Series instead. $\endgroup$ – bbgodfrey May 9 '17 at 23:29
  • $\begingroup$ In this case Series performs no better. $\endgroup$ – bbgodfrey May 10 '17 at 1:27
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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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