Preheat oven to 350°F (180°C).
Grease a 9x13-inch (23x33cm) rectangular pan with softened butter or shortening, or use baking spray. Line it with a large piece of parchment paper, covering the bottom and two long sides. The two short sides will be greased but not lined. You can also use an 8x12-inch (20x30 cm) pan and make a taller cake.
In a small bowl, combine ½ cup lime juice, ½ cup milk and 1 teaspoon vanilla extract. It will curdle and that's fine.
In a medium bowl, sift together 3 cups all-purpose flour, 1 tablespoon baking powder and ¼ teaspoon salt. Or have the ingredients measured and sift them directly over the cake mixture when prompted (my choice).
In a large bowl, beat 1 ¼ cups unsalted butter and 1 ¾ cup sugar for 3 minutes, until light. Add 1 teaspoon lime zest.
Add 5 eggs one at a time, beating well between additions.
Add the dry ingredients in 3 parts, alternating with the liquid mixture in 2 parts. That means you start and end with the flour mixture. You want to incorporate everything well until you have a smooth, lump-free batter, but don't overbeat at this point.
Pour the cake batter into the prepared pan, smooth the top, and bake for about 35-40 minutes, until golden brown and a cake tester or toothpick inserted in the center comes out dry.
Let the cake cool down completely on a wire rack.