In many areas of applications involving hypergeometric functions of one, two, and more variables, one requires the behaviors of these functions near the boundaries of the regions of convergence of the series defining them. Several results of this type have appeared in the literature [see, e.g., the first author, Bull. Cent. Res. Inst., Fukuoka Univ., Nat. Sci. 104, No.24, 13-31 (1988; Zbl 0644.33004)], some of these results were used in solving various boundary value problems involving the celebrated Euler Darboux equation and the study of certain operators of fractional calculus. The object of the present paper is to derive similar behaviors of the Appell double hypergeometry function \(F_ 4\) and the Lauricella triple hypergeometric functions \(F_ C\), \(F_ N\), and \(F_ R\) [see, for definitions, the second author and P. W. Karlsson, Multiple Gaussian hypergeometric series (1985; Zbl 0552.33001)].