Summary: Let \(\Omega\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq 2}\), and let \(u_p\in W_0^{1,p}(\Omega)\) denote the \(p\)-torsion function of \({\Omega}\), \(p >1\). It is observed that the value 1 for the Cheeger constant \(h(\Omega)\) is threshold with respect to the asymptotic behavior of \(u_{p}\), as \(p\to 1^+\), in the following sense: when \(h(\Omega) > 1\), one has \(\lim_{p\to 1^+} \left\|u_{p} \right\| _{L^\infty(\Omega)}=0\), and when \({h(\Omega) < 1}\), one has \(\lim_{p\to 1^+} \left\|u_p\right\| _{L^\infty(\Omega)}=\infty\). In the case \(h(\Omega)=1\), it is proved that \(\limsup_{p\to1^+}\left\| u_p\right\|_{L^\infty(\Omega)}<\infty\). For a radial annulus \(\Omega_{a,b}\), with inner radius \(a\) and outer radius \(b\), it is proved that \({\lim_{p\to 1^+} \left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \(h(\Omega_{a,b})=1\).