Newton's method is sensitive to an initial guess. It exhibits chaotic behavior that generates interesting fractal images. Most of them have either a finite number of attractors (attractive fixed points) or unbounded Julia sets. In this paper, we show that Newton's method for a family of equations exp(-alpha zeta + z/zeta - z) - 1 = 0 (for alpha > 0 and Absolute value of zeta = 1) has infinitely many attractors and a bounded Julia set. The dynamics of Newton's method for finding their roots are also visualized.