On the shape of graphs for n-dimensions

Question

As you all know, the graph of the function $x^2+y^2=1$ is a circle. Also, the graph of the limit as n approaches infinity of $x^n+y^n=1$ approaches a square. Will this be true for higher dimensions? Will the graph of $x^n+y^n+z^n=1$,or $x^n+y^n+z^n+t^n=1$ as n approaches infinity, be a cube of n dimensions?

Answer 1

Yes, it will. Given a number $x \lt 1,$ as $n$ increases $x^n$ gets smaller and smaller, so in three dimensions if you have $x^n+y^n+z^n=1$ one of the three variables must be very close to $1$ and the surface looks a lot like a cube.