Check with http://www.everymac.com which will list the maximum resolution supported by your Mac.
http://easyresapp.com/ has a great software called EasyRes, which once launched puts an icon by your clock in the upper right corner of the screen. This icon allows you to easily switch resolutions and frequencies. Some resolution and frequency switches are done on the LG's own on screen menu. Check with LG's support for your monitor's model number if you have difficulty reaching the right menu for on screen resolution.
Additionally your extended display resolution may be higher than the mirroring because the limited number of pixels of your screen on the Mac.