Interesting Limit (26)

For $x \in (-1,1), \displaystyle \lim_{n \to \infty}x^{2n}=0 \implies y=\pm 1$ . Since the equation is symetrical, this also holds when we swap $x$ and $y$ .

The resulting curve is a square of perimeter $\boxed{8}$ .

See What is the limit volume of this figure as n goes to infinity?

"This is not a requirement to solve this problem. If you like, then determining this limit will give you a major clue:

$\text{Assuming}\left[\left| x\right| \leq 1,\underset{n\to \infty }{\text{lim}}\ \underset{x\to 1}{\text{lim}} \ x^{2n}\right]$ "

$\lim_{n\rightarrow \infty}x^{2n}=\begin{cases} 0 & \text{for }|x|<1\\ 1 & \text{for }|x|=1\\ \infty & \text{for }|x|>1 \end{cases}$

The only way to make the equation true at $n\rightarrow \infty$ is for one of $\{|x|,|y|\}$ to be 1 and the other to be less than 1. The locus of points thus described is a square of perimeter $\boxed{8}$ .

By inspection, in the limit:

1) Neither $x$ nor $y$ can have a magnitude greater than $1$
2) $x$ and $y$ cannot both have a magnitude less than $1$
3) The only permissible combination is one variable at exactly $1$ , with the other variable having a magnitude of less than $1$
4) This results in a square, centered on the origin, with a side length of $2$ and a perimeter of $8$

It therefore appears that a square is related to a circle in a somewhat unexpected way, with both curves being limiting cases of a more general curve. The square is the "high $n$ " limit and the circle is the "low $n$ " limit

How about the curve length of $|x|^{\frac{m}{n}}+|y|^{\frac{m}{n}}=1$ where $m$ , $n$ are positive integers and $n \to \infty$ ?

As n gets larger and larger, the contours of x^(2n) + y^(2n) = 1 approach that of a SQUARE with a side length of 2, centered at (0,0)

When y is from interval (-1,1) the y^(2n) as n goes to infinity goes to zero, because even if it is 0,999999999 it must go to zero. So as y is from that interval x is either 1 or -1 and when y is either 1 or -1 x is from interval (-1,1) so when we imagine, how will the graph looks like, we will notice that it is a square with side lenght 2, so the answer is the perimeter of this square, which is 2*4=8.