Grease and line the base of an 18 x 28 cm (7 x 11 inch) roasting tin with nonstick baking paper. 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. Turn into the tin and level the surface.
Bake in a preheated oven, 180°C (350°F), Gas Mark 4, for about 20 minutes until well risen and firm to touch. Leave to cool in the tin for 10 minutes then transfer to a wire rack.
Make the lime frosting by beating the butter, icing sugar and half the lime rind and juice together to make a smooth light mixture. Turn the cake over so the top is uppermost, then spread with the frosting. Decorate with a sprinkling of the remaining lime rind, ready-to-eat dried fruits and coconut shavings. Cut into 20 squares to serve.