In　this　paper,　we　investigate　the　limits　of　Riemann　solutions　to　the　Euler　equations　of　compressible　fluid　flow　with　a　source　term　as　the　adiabatic　exponent　tends　to　one.　The　source　term　can　represent　friction　or　gravity　or　both　in　Engineering.　For　instance,　a　concrete　physical　model　is　a　model　of　gas　dynamics　in　a　gravitational　field　with　entropy　assumed　to　be　a　constant.　The　body　force　source　term　is　presented　if　there　is　some　external　force　acting　on　the　fluid.　The　force　assumed　here　is　the　gravity.　Different　from　the　homogeneous　equations,　the　Riemann　solutions　of　the　inhomogeneous　system　are　non　self-similar.　We　rigorously　proved　that,　as　the　adiabatic　exponent　tends　to　one,　any　two-shock　Riemann　solution　tends　to　a　delta　shock　solution　of　the　pressureless　Euler　system　with　a　Coulomb-like　friction　term,　and　the　intermediate　density　between　the　two　shocks　tends　to　a　weighted　δ-mesaure　which　forms　the　delta　shock;　while　any　two-rarefaction-wave　Riemann　solution　tends　to　a　two-contact-discontinuity　solution　of　the　pressureless　Euler　system　with　a　Coulomb-like　friction　term,　whose　intermediate　state　between　the　two　contact　discontinuities　is　a　vacuum　state.　Moreover,　we　also　give　some　numerical　simulations　to　confirm　the　theoretical　analysis.　©　2020,　Springer　Nature　B.V.