macbook air 6.2 mid 2013 runs out of memory on Yosemite. Safari running slow.

It's not a problem. Yosemite uses RAM differently. It keeps it active. Use your Activity Monitor, if the memory pressure is green, you're good to go.

Are you using any extensions in Safari?

Do other browsers work normally?

Please answer as many of the following questions as you can. You may already have answered some of them. In that case, there's no need to repeat the answers.

Back up all data before making any changes.

Have you restarted your router and your broadband device (if they're separate) since you first noticed the problem? If not, do that now and see whether there's any change.

If your browser is Safari, then from the Safari menu bar, select

Safari ▹ Preferences... ▹ Privacy ▹ Remove All Website Data

and confirm. If the Downloads button (with the icon of a downward-pointing arrow) is showing in the toolbar, click it and then click Clear in the box that appears. The download history will be removed. Any change?

If you're running OS X 10.9 or later, select the Advanced tab in the Preferences window and uncheck the box marked

Stop plug-ins to save power

Any change?

Quit and relaunch the browser. Any change?

Enable guest logins* and log in as Guest. Don't use the Safari-only “Guest User” login created by “Find My Mac.”

While logged in as Guest, you won’t have access to any of your documents or settings. Applications will behave as if you were running them for the first time. Don’t be alarmed by this behavior; it’s normal. If you need any passwords or other personal data in order to complete the test, memorize, print, or write them down before you begin.

Test while logged in as Guest. Same problem?

After testing, log out of the guest account and, in your own account, disable it if you wish. Any files you created in the guest account will be deleted automatically when you log out of it.

*Note: If you’ve activated “Find My Mac” or FileVault, then you can’t enable the Guest account. The “Guest User” login created by “Find My Mac” is not the same. Create a new account in which to test, and delete it, including its home folder, after testing.

Are any other web browsers installed, and are they the same? What about other Internet applications, such as iTunes and the App Store?

If other browsers and Internet applications are also affected, follow these instructions and test. Any change?

If Parental Controls is active for any user, please turn it off and test. Any change?

If only Safari is affected, launch the Activity Monitor application and enter "web" (without the quotes) in the search box. If a process named "Safari Web Content" is shown in red or is using more than about 5% of a CPU, select it and force it to quit by clicking the X or Quit Process button in the toolbar of the window. There may be more than one such process. Any improvement?

Follow the instructions in this support article. Any change?

Open the iCloud preference pane and uncheck the box marked Photos, if it's checked. Any change?

Are there any other devices on the same network that can browse the Web, and are they affected?

If you can test Safari on another network, is it the same there?

If you connect to your router with Wi-Fi and you can also connect with Ethernet, do that and turn off Wi-Fi. Any difference?

RAM is a non-issue. Is Safari still slower than you think it should be? Did you follow the instructions to disable prefetching?

Please read this whole message before doing anything.

This procedure is a diagnostic test. It’s unlikely to solve your problem. Don’t be disappointed when you find that nothing has changed after you complete it.

The purpose of the test is to determine whether the problem is caused by third-party software that loads automatically at startup or login, by a peripheral device, by a font conflict, or by corruption of the file system or of certain system caches.

Disconnect all wired peripherals except those needed for the test, and remove all aftermarket expansion cards, if applicable. Start up in safe mode and log in to the account with the problem. You must hold down the shift key twice: once when you turn on the computer, and again when you log in.

Note: If FileVault is enabled in OS X 10.9 or earlier, or if a firmware password is set, or if the startup volume is a software RAID, you can’t do this. Ask for further instructions.

Safe mode is much slower to start up and run than normal, with limited graphics performance, and some things won’t work at all, including sound output and Wi-Fi on certain models. The next normal startup may also be somewhat slow.

The login screen appears even if you usually login automatically. You must know your login password in order to log in. If you’ve forgotten the password, you will need to reset it before you begin.

Test while in safe mode. Same problem?

After testing, restart as usual (not in safe mode) and verify that you still have the problem. Post the results of the test.