I have a nikon AF nikkor lens 75-300mm.

You did not say which camera body you are using, but you probably have three different auto-focus modes on you camera. You might be using the wrong one.

The modes are as follows:
AF-A Mode: Camera automatically selects single-servo autofocus when AF-A subject is stationary, continuous-servo autofocus when subject is moving. Shutter can only be released if camera is able to focus. AF-S Mode: For stationary subjects. Focus locks when shutter-release button AF-S is pressed halfway. Shutter can only be released when in-focus indicator is displayed.
AF-C Mode: For moving subjects. Camera focuses continuously while AF-C shutter-release button is pressed halfway. Photographs can be taken even when in-focus indicator is not displayed.

The D5000 does not have a focus motor in the body, relying on the lens to have one. The old 70-300mm G lens does not have a focus motor, relying on the camera to have one. Thus, this combination will not autofocus.
The newer 70-300mm AF-S G lens, as well as the even newer 70-300mm VR AF-S lens, do have the autofocus motor and thus will autofocus on the D5000.

If your lens is 8 months old, it is still in warranty. Contact Nikon for return instructions.

Hi,
Sorry to hear about your lens. If you are able to autofocus and take pictures with any other lens, then it means that your problem lies with the kit lens. If you can take picture in manual mode, but not in autofocus mode, then the AF motor in the lens in not functioning. I had the same thing happen to my 50mm 1.4 lens.
Depending on your location, there are Nikon repair centers in Melville, NY and Torrance, CA.
Hope this helps,
Andrew

problem in lens censer try to set the censer or replace it

Press the AF button on the top of the camera (to the right of the window) to select an AF mode & make sure the M/A (Manual / Auto focus) switch on the lens set to "A"

More info below:

http://www.kenrockwell.com/nikon/d80/af-settings.htm

Seems as if you're snapping the shutter before the lens has focused. Have you tried manual focus or single-server auto focus?

Please check your focus settings in the camera menu. Play around with the focus point from Single to 11 points. Also, try Auto mode as a quick way to test if the focus is working

Is the switch on the side of the lens set to manual or auto? Sometimes it's the little things like this that we don't notice.

Which camera are you using? If the body has AF/MF switch, put that also to MF.If this fails to help, take out the lens and try to change it to MF.