Autofocus stops after heavy usage

Tamron makes their lenses with a variety of mounts. A lens with a Canon mount, for example, will not fit a Nikon. Assuming the lens has a Nikon mount then yes, it is compatible with the D60. However, the D60 lacks an autofocus motor, relying on the lens to have one. Some Tamron lenses have such a motor, some do not. Such lenses will not autofocus with the D60, but everything else still works, including manual focus.

This lens lacks the autofocus motor, so it will only autofocus on cameras with the motor. This lens will autofocus on every Nikon digital SLR except for the D40, D40x, D60, D3000, D3100, D5000, and D5100. It will autofocus with the D1 family, D2 family, D3 family, D50, D70, D70s, D80, D90, D100, D200, D300, D300s, and D700 (I hope I haven't left any out).

Sorry, but the D40x lacks an autofocus motor, depending on the lens to have one. Nikon calls such lenses AF-S. This particular lens does not have such a motor, relying on the camera to have one. The only 50mm lens currently available with an autofocus motor is this one:
http://www.nikonusa.com/Nikon-Products/Product/Camera-Lenses/2180/AF-S-NIKKOR-50mm-f%252F1.4G.html

Earlier this week the Nikon web site listed a new 50mm F1.8 lens with an autofocus motor, but then it was removed. Hopefully they will officially announce this lens soon.

The D40x only autofocuses with the latest AF-S (and old professional AF-I) lenses. Most traditional (screw-type) AF lenses from 1986 through today will not autofocus on the D40x. If your lens says "AF-S," you're fine.
Most lenses sold today for the D40x are AF-S, and so is the included 18-55mm kit lens. Many non-Nikon lenses from Tamron, Tokina, Quantaray, Sigma and anyone other than Nikon still won't autofocus on the D40x. Try before you buy, but honestly, there isn't much the included 18-55mm lens won't do.
Autofocus systems are fast, but not usually instantaneous. You have to hold the shutter down halfway as you compose the first shot so the AF system can focus and lock, and then the camera fires instantaneously when you press the shutter the rest of the way. See Preventing Shutter Delay for more.

That lens doesn't have an internal focusing motor and thus will not autofocus with the D40x. Nothing is broken.

All Sigma:
50-500mm f/4-6.3 DG
50-150mm f/2.8 EX DC HSM
70-200mm f/2.8 DG MACRO
80-400mm f/4.5-5.6
100-300mm f/4 DG
120-300mm f/2.8 DG

We tried the same lens on another Nikon (D90) and auto focus did not work.
We mounted a new set of lens on the original D40x, and it worked.
Turns out the 2 year Nikon lens (18mm-135mm) was bad. Nikon lens have a 5 year warrantee. The lens will be sent back to Nikon with no service or shipping charge.

Thanks,
K

Bad lens. But go to camera shop with lens just to confirm.

Try removing the lens and then reloading it.   Hope this works.

Autofocus function on the D40 only supports lenses with the
AF-S feature, which have an autofocus motor built into the lens,
instead of using an autofocus motor drive built into the camera.

The Tamron lens you have, does not have a built in motor and
the autofocus function relies on the motor drive in the camera.

It will not work with the D40 or D60, but it will work with other
Nikon digital SLRs, such as the D80. Need to get an AF-S
type lens or upgrade to different Nikon digital SLR. Unless
you do this, you will have to manually focus the Tamron lens.

All of the Nikon DX (for digital) series lenses are also AF-S
type, so they will work with the D40, which is designed as
an entry level digital SLR, therefore it is intended primarily
for use with DX lenses, which are typically sold in a kit with
the D40. Most common one is Nikkor 18-55mm DX AF-S.

Other Nikon AF-S lenses made for film cameras (FX type)
will also work with the D40, but these tend to be expensive
professional models. DX series lenses tend to be more
affordable. You might consider the 18-200mm DX AF-S
as an alternative to the Tamron, but these are not cheap.