@DanielFischer We have the cardioid
$z(t)=(1+ \cos t) (\sin t-i \cos t), 0 \leq t \leq 2\pi$
The lngth of the curve is given by $\int_a^b |z'(t)| dt$
There is a subsequence $ \frac{k^{\sin k}}{\sqrt{k}}$ such that $ \frac{k^{\sin k}}{\sqrt{k}} \to 0$ and a subsequence $ \frac{k^{\sin k}}{\sqrt{k}}$ such that $ \frac{k^{\sin k}}{\sqrt{k}} \to \sqrt{n} \to =\infty$
How can we find the perimeter of the cardioid curve?