In this paper, we study the Cauchy problem of the isothermal system in a general nozzle with space-dependent friction α(x). First, by using the maximum principle, we obtain the uniform bound ρδ,ε,τ≤M, |mδ,ε,τ|≤M, independent of the time, of the viscosity-flux approximation solutions; Second, by using the compensated compactness method coupled with the convergence framework given in [5], we prove that the limit, (ρ,m) of (ρδ,ε,τ,mδ,ε,τ), as ε,δ,τ go to zero, is a uniformly bounded entropy solution.