Very simple, no library required:
var date = new Date();
var firstDay = new Date(date.getFullYear(), date.getMonth(), 1);
var lastDay = new Date(date.getFullYear(), date.getMonth() + 1, 0);
or you might prefer:
var date = new Date(), y = date.getFullYear(), m = date.getMonth();
var firstDay = new Date(y, m, 1);
var lastDay = new Date(y, m + 1, 0);
Some browsers will treat two digit years as being in the 20th century, so that:
new Date(14, 0, 1);
gives 1 January, 1914. To avoid that, create a Date then set its values using setFullYear:
var date = new Date();
date.setFullYear(14, 0, 1); // 1 January, 14