Preheat your oven to 350°F (175°C). Grease a 9x9 inch baking pan.
In a mixing bowl, whisk together the melted butter, brown sugar, and granulated sugar until smooth.
Add the eggs and vanilla extract, and whisk until well combined.
In another bowl, mix the flour, baking powder, and salt. Gradually add this to the wet ingredients, mixing until just combined.
Fold in the shredded coconut, lime zest, and lime juice.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 25 minutes or until a toothpick inserted into the center comes out clean.
Allow to cool in the pan before cutting into squares and serving.