Preheat the oven to 350 F or 176C.
Line muffin/cupcake pan with liners.
In a bowl combine all purpose flour, sugar, baking powder, salt and keep side.
In a separate bowl take blackberries sprinkle some flour (from above step, about ½ - 1 cup) to coat the blackberries. Generously coat the berries and this will help the berries from sinking to the bottom of the cupcake tin when baking. This is an optional step and check notes below.
In a large bowl mix oil, eggs, lime zest, milk and vanilla.
Stir in the dry ingredients into wet ingredients. Then gently fold or stir in blackberries.
Fill lined cupcake/muffin cups with batter ⅔ way and bake for 20-25 minutes or until toothpick inserted comes out clean. Mine took 23 ½ minutes and for mini cupcakes, 17 minutes.