  We study incompressible surfaces constructed by Culler-Shalen theory in the
context of twisted Alexander polynomials. For a $1$st cohomology class of a
$3$-manifold the coefficients of twisted Alexander polynomials induce regular
functions on the $SL_2(\mathbb{C})$-character variety. We prove that if an
ideal point gives a Thurston norm minimizing non-separating surface dual to the
cohomology class, then the regular function of the highest degree has a finite
value at the ideal point.
