Editing SR/Ptot QuarticSolution (section)

===Limiting Regimes===
We can immediately see that this solution makes sense in the present context.  In order for the temperature &#8212; that is, <math>z</math> &#8212; to be real and nonnegative, the function <math>\phi(\lambda)</math> must be greater than or equal to <math>2</math>.  This limiting value occurs when the dimensionless parameter, <math>\lambda = 0</math>.  The constraint <math>\lambda \ge 0</math> is satisfied as long as the three coefficients of the quartic equation are real and nonnegative, which is certainly true for our specific problem.

Looking at the limiting functional behavior of our solution, we see that when <math>~0 \le \lambda \ll 1</math>,
<div align="center">
<math>
\phi \approx 2 + \frac{3}{2^{2/3}}\lambda 
</math><br />

<math>~
\Rightarrow ~~~~~ z \approx \frac{3\lambda}{2^2} \biggl( \frac{a_1}{2^2a_4} \biggr)^{1/3} = \frac{a_0}{a_1} .
</math>
</div>


We see, as well, that when <math>\lambda \gg 1</math>,
<div align="center">
<math>
\phi \approx \biggl(\frac{3\lambda}{2^2} \biggr)^{3/2}
</math><br />

<math>
\Rightarrow ~~~~~ z \approx \biggl(\frac{3\lambda}{2^2} \biggr)^{1/4} \biggl( \frac{a_1}{2^2a_4} \biggr)^{1/3} = \biggl( \frac{a_0}{a_4} \biggr)^{1/4} .
</math>
</div>