HMMT 2001 K1

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 $\lim _{n \rightarrow \infty} L_{n}$.