Auto focus quit

I would consider focus shift when zooming normal. Most small digital camera zoom lenses do that. If your auto-focus is slow, all I can suggest is patience or a newer/better camera.

Sounds like the autofocus motor is having issues, I'd suggest contacting Sigma.

I do not think your lens is out of order.
Sometimes this happens.
If your lens is not out of order then lets think the problem is another thing according to the user.
Autofocusing is great but under some conditions such as below you may consider.
Shooting in dimly lit environments can be difficult for some cameras and lenses when it comes to focusing.
You’ll know when your camera is struggling in Auto mode when every time you go to take a shot the lens will whirl from one end of it’s focusing options to the other and back again before deciding on where to focus.
This can really lengthen your shooting process and make taking quick candid shots quite frustrating.
Switch to manual mode and you can quickly find your focusing point and get the shot you’re after.

try cleaning auto feed sensors

google its user manual

changing lens from stock might be the issue
or 8 years auto focus sensor may be stuffed

Are you sure you did not activate "Manual Focus"?
Near the shutter button, there's a "Focus" button. maybe you pressed it accidentally and now you're in "Manual Focus" mode.

If so, press it once more to get "Multipoint" mode back to life...

And read the camera manual. It's a goot place to look for this kind of answers.

I had the same problem. My issue was with all my lenses. I solved it by going to the menu and selecting the custom setting menu (pencil icon). Then from the right menu select the AE-L/AF-L and press OK. Then make sure it is set on AE/AF Lock. Presto...all my lenses worked on auto-focus.

Sounds like you need to get your camera fixed by Nikon.

However, you say the auto focus stopped working, you didn't switch to a lens that uses a mechanical focus linkage do you? The D40 only supports lenses with AF-S or AF-I type autofocus drives. AF-D and AF-G types without AF-S or AF-I don't autofocus.

Also, check that the A/M switch on the lens is in the correct position.

Manual focusing is quite difficult to do accurately with an AF camera. The screen inside the camera is not optimized for this. If you are sure you are focusing accurately but the pictures you take are still out of focus, the mirror may need adjusting. I can provide further info on this if you require, but if you're within warranty, i'd recommend that route first.

I had the same problem a few moths ago when I tried to use a Quanta lens. Go into the second tools menu, select clear settings, then select Clear all Custom Functions. It worked for me I hope it works for you.

I was first going to also suggest a reset. Does your manual focus function correctly from near to far(turning the focus ring in manual) and does the wide to tele motion function fully and smoothly? (lens adjustability without restriction) Was your camera exposed to extreme heat or elements?(beach sand or water of any kind) It may mean electronics only or a mechanical restriction causing an electronic malfunction. A last resort may mean your closer documenting when or how frequent it happens and maybe ultimately a trip (for your camera) to Sony. Get a price quote for repair since you might strike a happy medium of using the camera differently (manually focussing) without exceeding the value of the camera for repair costs.