1. A sequence of ants walk from $(0,0)$ to $(1,0)$ in the plane. The $n$th ant walks along $n$ semicircles of radius $\frac{1}{n}$ with diameters lying along the line from $(0,0)$ to $(1,0)$. Let $L_{n}$ be the length of the path walked by the $n$th ant. Compute $\lim L_{n}$.
Solution: A semicircle of radius $\frac{1}{n}$ has length $\frac{1}{2} \pi\left(\frac{2}{n}\right)=\frac{\pi}{n}$, so $n$ such semicircles have total length $\pi$.