We derive asymptotic formulas for the coefficients of certain classes of weakly holomorphic Jacobi forms and weakly holomorphic modular forms (not necessarily of integral weight) without using the circle method. Then two applications of these formulas are given. Namely, we estimate the growth of the Fourier coefficients of two important weak Jacobi forms of index 1 and nonpositive weights and obtain an asymptotic formula for the Fourier coefficients of the modular functions θk/ηl for all integers k, l ≥ 1, where θ is the weight 1/2 modular form and η is the Dedekind eta function.