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.