In a mixing bowl, cream together the butter and sugar.
Add to room temp eggs, one at a time. Continue mixing until combined. Then add the vanilla, lime juice and oil, and blend in for one minute.
Slowly add dry to wet in no more than 3 batches, alternating with the milk, and ending with the flour, being careful to not overmix the cake batter.
Fold in the lime zest by hand with a spatula.
Bake in a greased and floured bundt pan at 350 degrees for 35 minutes.
Instead of whole milk, buttermilk can be substituted if preferred, making for an even moister and richer cake.