I think you will need to maintain the mappings yourself (you could XHR a physical file and get it, but I doubt you want to do that)...
(function() {
    var extToMimes = {
       'img': 'image/jpeg',
       'png': 'image/png',
       ...
    }
    window.getMimeByExt = function(ext) {
        if (extToMimes.hasOwnProperty(ext)) {
           return extToMimes[ext];
        }
        return false;
    }
})();
jsFiddle.
This will return false if it doesn't exist in your mapping. If you like, you could just return the extToMimes[ext], which will give undefined if it doesn't exist. It also won't accidentally get things up the prototype chain.
There is also the option of accessing it if it is a file input.
var mime = $('input[type="file"]').prop('files')[0].file;
Keep in mind the extension to mime type makes no guarantees about the actual file itself; you'd need to parse it server side to determine what kind of file it really is.