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A well-known example for this type of curve is the "Koch Snowflake" curve. This is a fractal curve that is continuously expanded, yet it remains within a limited area. To understand better, let us examine the construction process of the Koch Snowflake curve.

The construction of the Koch Snowflake curve begins with an equilateral triangle. At each iteration step, we add a smaller equilateral triangle on each side of the current figure. The new triangles are added in such a way that they share one side with the current figure and the two other sides are outside of the figure. This process is continued indefinitely.

Let \(a\) be the length of the side of the initial equilateral triangle. After the first iteration, four sides of the same length as the starting triangle will form the new curve. Thus, the total length of the curve after the first iteration, \(L_1\), is: \[L_1 = 4a\] In the second iteration, each of the four sides will be further divided, and smaller triangles will be added, resulting in \(4\) new sides of length \(\frac{a}{3}\) for each of the initial four sides. So, after the second iteration, we have 4 times 4 sides of length \(\frac{a}{3}\), which means the total length, \(L_2\) is: \[L_2 = 4 \cdot 4 \cdot \frac{a}{3}\] In general, after \(n\) iterations, the total length \(L_n\) will be: \[L_n = 4^n \cdot \frac{a}{3^{n-1}}\]

As the number of iterations, \(n\), approaches infinity, the term \(4^n\) in the expression for the total length becomes infinitely large, as well: \[lim_{n \to \infty} 4^n = \infty\] Since the term \(\frac{a}{3^{n-1}}\) remains positive during the iterations, the product of an infinitely large term and a positive term will also be infinitely large. Therefore, the arc length, \(L_n\), approaches infinity as \(n\) approaches infinity: \[lim_{n \to \infty} L_n = \infty\]

It has been shown that a curve of infinite arc length can exist within a limited area in the 鈩澛� (real plane). The example of the Koch Snowflake curve demonstrates this property, as its arc length approaches infinity while still being confined to a limited area.