The unstable circular photon orbits around black holes encode rich information about the geometry and dynamics of spacetime. In the high-frequency limit, the eikonal quasinormal modes (QNM) of a black hole are intimately connected to the exponential divergence of these circular photon orbits. We present an alternative geometric derivation of this correspondence by employing the Penrose limit, which captures the physics near the photon ring. Remarkably, the Lyapunov exponents, orbital frequencies, and QNM spectrum all emerge from a single Hamiltonian defined within the Penrose limit geometry. This unified framework also raises the intriguing possibility that Lyapunov exponents are subject to upper bounds, potentially constrained by the spacetime geometry properties.