Heat the butter, sugar and syrup gently in a saucepan, stirring until melted.
Mix the dry ingredients together in a mixing bowl then stir in the melted butter mixture and beat together until smooth. Stir in the eggs then the chopped pineapple, reserving a few pieces for decoration if liked.
Pour the mixture into an 18 x 28 cm (7 x 11 inch) roasting tin, greased and base-lined with oiled greaseproof paper, and spread the surface level.
Bake in a preheated oven, 180°C (350°F), Gas Mark 4, for about 20 minutes until well risen and the cake springs back when gently pressed with a fingertip. Leave to cool in the tin for 10 minutes then loosen the edges, turn out on to a wire rack and peel off the lining paper.
Make the lime frosting. Beat the butter, icing sugar, half the lime rind and juice together to a smooth fluffy mixture. Turn the cake over so the top is uppermost then spread with the lime frosting. Decorate with a sprinkling of the remaining lime rind, the ready-to-eat dried fruits and coconut shavings. Store in an airtight tin for up to 2 days. Cut into 20 pieces to serve.