By taking an appropriate limit, we obtain the Bessel functions related to root systems as limits of Heckman-Opdam hypergeometric functions. A more general class of Bessel functions is also investigated, which we shall call the $\Theta$-Bessel functions. Explicit formulas for the $\Theta$-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of non-compact causal symmetric spaces, were an integral representation for these special functions is shown.