In the bowl of a stand mixer fitted with the paddle attachment, beat the sugar, butter, and shortening on high for 1 minute. While that is beating, in a small bowl, combine the flour, baking powder, and salt. Once the butter mixture is combined, add in the eggs, sour cream, milk, and lime juice. Mix for another minute. Add in the lime zest and stir. Add in the flour mixture and stir to combine until the flour is integrated.
3 cups all-purpose flour, 1 teaspoon baking powder, ¾ teaspoon kosher salt, 2½ cups granulated sugar, ¾ cup unsalted butter, ¾ cup vegetable shortening, 4 large eggs, ¾ cup sour cream, ½ cup milk, ⅓ cup lime juice, 4 teaspoons lime zest