Summary: For a complex simple simply connected Lie group \(G\), and a compact Riemann surface \(C\), we consider two sorts of families of flat \(G\)-connections over \(C\). Each family is determined by a point \(\mathbf{u}\) of the base of Hitchin’s integrable system for \((G,C)\). One family \(\nabla_{\hbar ,\mathbf{u}}\) consists of \(G\)-opers, and depends on \(\hbar \in \mathbb{C}^\times \). The other family \(\nabla_{R, \zeta,\mathbf{u}}\) is built from solutions of Hitchin’s equations, and depends on \(\zeta \in \mathbb{C}^\times, R \in \mathbb{R}^+\). We show that in the scaling limit \(R \to 0, \zeta = \hbar R\), we have \(\nabla_{R,\zeta,\mathbf{u}} \to \nabla_{\hbar,\mathbf{u}} \). This establishes and generalizes a conjecture formulated by Gaiotto.