I assume it's \lim_{y\to \infty} (g(y)ln(ln(y)))/(ln(y))=1. Note that when y = x^x, y\to\infty is the same as x\to\infty, so \lim_{y\to\infty} gives the same result as \lim_{x\to\infty}. Express ln(y) and ln(ln(y)) through x and replace g(y) by x; then you will have a limit of some expression of x only.