In a medium saucepan, melt 1 tablespoon of the butter over medium-high heat. Add the rice and sauté until the rice begins to brown a little, 2 to 3 minutes. Add the broth and bring to a boil over high heat. Reduce the heat to medium-low, cover, and simmer until tender, about 15 minutes. Fluff the rice with a fork.
Meanwhile, in a small glass bowl, melt the remaining 3 tablespoons butter in the microwave, about 30 seconds. Stir in the lime zest and juice.
Pour the lime butter over the hot rice, add the cilantro, and stir until combined. Season with the salt and pepper.