Answer
$\lim\limits_{n \to \infty} A_n=1$
This implies that the given region approximates a square of side length $1$.
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx\\ = \int_{0}^{1} [x^{1/n}-x^n] \ dx \\= [\dfrac{nx^{1+n/n}}{1+n}-\dfrac{x^{n+1}}{n+1}]_{0}^{1} \\=\dfrac{n}{1+n}- \dfrac{1}{n+1} \\=\dfrac{n-1}{1+n}$
Now, $\lim\limits_{n \to \infty} A_n=\lim\limits_{n \to \infty}\dfrac{n-1}{1+n}=1$
This implies that the given region approximates a square of side length $1$.
