The graph of $x^{n}+y^{n}=r^{n}$ for sufficiently large $n$

Solution 1:

You have just rediscovered the max-norm.

More precisely, you have noted that as $p$ becomes large, the unit circle in the $l_p$ norm looks similar and similar to the one of the $l_\infty$ norm.

Solution 2:

Sometimes it's called a superellipse - see, e.g., http://en.wikipedia.org/wiki/Superellipse

Solution 3:

By symmetry, you can consider the equation $y^n+x^n=r^n$ for $0 \leq x \leq r$. Rewrite as $$ y(x) = \sqrt[n]{{r^n - x^n }} = \sqrt[n]{{r^n - r^n \bigg(\frac{x}{r}\bigg)^n }} = r\sqrt[n]{{1 - \bigg(\frac{x}{r}\bigg)^n }}, $$ for $0 \leq x \leq r$. This shows that $y$ is strictly decreasing from $r$ to $0$ as $x$ varies from $0$ to $r$, respectively, and that the sequence of functions $y(x) = y_n (x)$ converges pointwise, as $n \to \infty$, to the function $f$ defined by $f(x)=r$ if $0 \leq x < r$ and $f(r)=0$; moreover, the convergence to $f$ is uniform for $x \in [0,a]$, for any $0 < a <r$ (but not for $x \in [0,r]$, since $y(r)=0$). This accounts for the square shape.