I have a problem on a homework assignment. I will write the question and then what I have so far. I just want some guidance on what I am doing wrong (or right). Any help would be greatly appreciated:

In these exercises we study the equations of motion of a perfect fluid. These are the conservation laws:

\begin{eqnarray} \nabla_{\mu}T^{\mu\nu} = 0 \space (1) \\ \nabla_{\mu}I^{\mu} = 0 \space (2) \end{eqnarray}

Where $T^{\mu\nu}$ is the energy-momentum-stress tensor, and $I^{\mu}$ the particle current: \begin{eqnarray} T^{\mu\nu} = \rho{u}^{\mu}u^{\nu} + p((g^{-1})^{\mu\nu} + u^{\mu}u^{\nu}) \space (3) \\ I^{\mu} = nu^{\mu} \space (4) \end{eqnarray}

Here $\rho$ is the energy density, $u^{\mu}$ the material velocity (a future-directed unit time-like vectorfield), and p denotes the pressure while n is the [article density.

Exercise 1 [Nonrelatvisitic limit of the conservation laws]: Let us first study the quasi-Newtonian limit of the conservation laws. We consider (1) and (2) in Minkowski space and assume that with respect to a system of stationary observers: \begin{eqnarray} u^{0} = \frac{1}{\sqrt{1-\frac{|v|^2}{c^2}}}, u^i = \frac{\frac{v^i}{c}}{\sqrt{1-\frac{|v|^2}{c^2}}} \space (5) \end{eqnarray}

Moreover, we assume that \begin{eqnarray} \rho = \mu{c^2} + h \space (6) \end{eqnarray}

where $\mu$ is the mass density and $h$ is the internal energy density.

(i) Derive in the nonrelativistic limit $c\rightarrow\infty$ the conservation of mass law, \begin{eqnarray} \frac{\partial\mu}{\partial{t}} + \nabla\cdot(\mu{v}) = 0. \space (7) \end{eqnarray} Hints: Express the coponents of (3) explicitly in orders of c using (5) and (6), and derive the conservation law from the 0-component of (1).

So I have done a little work on this but I don't see how I can derive (7). Maybe there is some math trick I need to use or maybe this is a lot easier than I am making it out to be. So, I put (5) and (6) into (3) and I get the following:

\begin{eqnarray} T^{\mu\nu} = (p + \mu{c^2} + h)\frac{v^{\mu}v^{\nu}}{c^2-|v|^2} + p(g^{-1})^{\mu\nu} \end{eqnarray} I am not sure what I am supposed to do from this point. I am not sure if I derived this properly and am I supposed to take the derivative of this in terms of $t$ and what do they mean by expressing things as orders of $c$. Thank you and have a great day!