Produce a sketch of $|x|^n + |y|^n =1$ for each $n \in \{1, 2, 1000\}.$

Try to consider the case when both $x$ and $y$ are positive, for $n=1.$

Can the behaviour in the first quadrant be reproduced for the other three quadrants, given we take absolute values of both $x$ and $y?$

What is the formula for the distance from a given point to the origin? Does it look familiar?

How small is $0.1^{1000}?$ What about $0.5^{1000}$ or even $0.99^{1000}?$

The equation $|x|^n+|y|^n=1$ will yield a symmetric shape with respect to both $x$ and $y$ axes (why?). Thus we can focus only on the first quadrant where $x$ and $y$ are both positive i.e. where $x^n+y^n=1,$ and then reflect the curve onto the other three quadrants.

For $n=1,$ we have $y=1-x.$ Reflecting this onto other quadrants forms a diamond with corners at $(\pm1,0)$ and $(0,\pm1).$

For $n=2,$ we obtain an equation of the unit circle. This is true, because $x^2+y^2$ is the formula for the squared distance from a given point to the origin.

For $n=1000,$ think about what happens to any number strictly less than 1 when raised to a very big power. It tends to 0. Now consider all values of $x$ that are sufficiently less than 1, even up to 0.999; these will cause $x^n$ to decrease sharply towards $0,$ and $y^n$ to raise sharply towards $1,$ for all such $x.$ Thus the graph tends to become a square.