Fatou's Theorem states that a bounded analytic function in the unit disk admits radial limits almost everywhere on the unit circleCIf we relax the condition of boundedness offin the hypothesis of Fatou's theorem to unboundedness of the maximum modulus functionMf(r) such thatMfr<q(r),whereq(r) has a slow growth to ∞ we can only hope to obtain asymptotic values instead of radial limits at points ofC. The MacLane classAof analytic functions is the class of non constant analytic functions in the unit diskDthat have asymptotic values at a dense subset of the unit circleC. The main purpose of this paper is to consider a question posed by MacLane [M2] in 1970 on whether functions inAwith an arc tract have branch points. We answer this question positively in the special cases, where the finite asymptotic values are bounded, or where infinity is linearly accessible at least at two points of the end of the tract