In the two definitions of divergence，the flux-density divergence provides the physical definition of divergence, whereas the nabla-operator divergence serves as a mathematical tool for calculation and derivation. Using examples derived from Coulomb’s law of electrostatics, and based on the theory of limits, this paper reveals that the flux-density divergence and the nabla-operator divergence are non-equivalent. The differential form of Gauss’s flux theorem in Maxwell’s equations does not hold. Furthermore, keeping or discarding relatively higher-order infinitesimals yields different results of the flux-density divergence, which contradicts the theory of limits in calculus. This paper proposes the effective order infinitesimal theorem to enhance the theory of limits. In the expression of the flux-density divergence, the effective order of its infinitesimal variables is 3. However, during the mathematical derivation of the nabla-operator divergence from the flux-density divergence, only 2nd-order infinitesimal variables are retained. Gauss’s flux theorem in differential form does not hold, and the theory of limits in calculus exhibits deficiencies. Physics and mathematics are undergoing a revolutionary scientific transformation.