If we move an object a very tiny distance h away from the earth's surface, ΔU = U(R_E + h) − U(R_E). If we use U(r) = −GM_Em/r, then we can get a common denominator to get an h dependence of [R_E - (R_E + h)]/((R_E + h)R_E) and if we now use the fact that h is negligible compared to the radius of the earth, we wind up with ΔU ≃ (GM_Em h)/R_E². But we know that g = GM_E/R_E² so we find that if h is vanishingly small compared to the planet radius, we wind up with the crude approximation ΔU ≃ mgh. It should never be used for any homework or example in this chapter.