Both sleep and hibernate are perfected in Windows 7 and should run fine as long as you have the latest Display driver
installed in Device manager, and have your Power options set to Sleep and then Hibernate.
Check for newer Display driver by running Windows Updates after enabling Automatically deliver drivers via Windows Update (Step 3)
and then running all Important and Optional Updates until no more are offered. Set future Updates to Automatic.
If no newer driver is delivered, check the Asus Support Downloads webpage for your model PC to install the latest: Driver Install - Device Manager
Be aware that the factory preinstalled Windows 7 is a corrupt install to begin with, loaded with bloatware and useless duplicate utilties which have better versions built into Windows 7. This is why most tech enthusiasts choose to Clean Reinstall - Factory OEM Windows 7
. At the minimum I'd Clean Up Factory Bloatware