In this paper we contrast two approaches for proving the validity of relaxation limits <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \rightarrow \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of systems of balance laws <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline plus f left-parenthesis u right-parenthesis Subscript x Baseline equals alpha g left-parenthesis u right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="1em"/> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} u_t +{f(u)}_x = \alpha g(u) \quad . \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> In one approach this is proven under some suitable stability condition; in the other approach, one adds artificial viscosity to the system <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline plus f left-parenthesis u right-parenthesis Subscript x Baseline equals alpha g left-parenthesis u right-parenthesis plus epsilon u Subscript x x"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>ϵ</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} u_t +{f(u)}_x = \alpha g(u) + \epsilon u_{xx} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> and lets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \rightarrow \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\epsilon \rightarrow 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> together with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M alpha less-than-or-equal-to epsilon"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M \alpha \leq \epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a suitable large constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We illustrate the usefulness of this latter approach by proving the convergence of a relaxation limit for a system of mixed type, where a subcharacteristic condition is not available.