This picks up #14599, which the inactivity bot closed while I was away — my fault for going quiet. The
bot suggested opening a new PR, so here it is, rewritten rather than restored.

The gap: a background task that raises cannot change a response that has already been sent, and the
exception never reaches the path operation function. From the outside a failed task looks exactly like
a successful one. #2505, #2604 and #3589 all come back to that, and the tutorial does not mention it.

What changed since the earlier attempt, after reading how the rest of the docs are written:

• the example moved into docs_src/background_tasks/tutorial003_py310.py and is referenced with
  {* ... *}, so it is covered by a test like the other examples rather than being an inline block
• /// tip instead of !!! tip, which is what the other 37 files in the tutorial use
• the heading is "Handle errors in the task", matching the sentence-case style of its neighbours
• the section is shorter, and points at the existing Caveat rather than repeating it

Two tests in tests/test_tutorial/test_background_tasks/test_tutorial003.py: the example runs, and a
task that raises still returns 200 while the error reaches the log — which is the behaviour the section
is actually describing. 5 passed in that directory.

Happy to trim it further if it is more than you want in a tutorial page.