We prove existence of scaling limits of sequences of functions defined by the recursion relation w(n+1)(1) (x) = -w(n)(x)(2). which is a successive approximation to w(1) (x) = -w(x)(2), a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n -> infinity in an asymptotically conformal way, w(n)(x) asymptotic to q(n)w(-)(q(n)-x), for a sequence of numbers {q(n)} and a function w(-). We also discuss implication of the results in terms of random sequential bisections of a rod.