Docs: Tail Sampling Caveat: Spans starting after root ended, e.g. background tasks (#1033)

Author: alexmojaki

docs/how-to-guides/sampling.md

@@ -202,6 +202,66 @@ If you start a trace with the Logfire SDK with tail sampling, and then propagate
 spans generated by the SDK may be discarded, while the spans generated by the other process may be included, leading to
 an incomplete trace.
 
+### Spans starting after root ended, e.g. background tasks
+
+When the root span of a trace ends, if the trace doesn't meet the tail sampling criteria, all spans in the trace are
+discarded. If you start a new span in that trace (i.e. as a descendant of the root span) after the root span has ended,
+the new span will always be included anyway, and its parent will be missing. This is because the tail sampling mechanism
+only keeps track of active traces to save memory. This is similar to the distributed tracing case above.
+
+Here's an example with a FastAPI background task which starts after the root span corresponding to the request has
+ended:
+
+```python
+import uvicorn
+from fastapi import BackgroundTasks, FastAPI
+
+import logfire
+
+app = FastAPI()
+
+logfire.configure(
+    sampling=logfire.SamplingOptions.level_or_duration(
+        duration_threshold=0.1,
+    ),
+)
+logfire.instrument_fastapi(app)
+
+
+async def background_task():
+    # This will be included even if the root span was excluded.
+    logfire.info('background')
+
+
+@app.get('/')
+async def index(background_tasks: BackgroundTasks):
+    # Uncomment to prevent request span from being sampled out.
+    # await asyncio.sleep(0.2)
+
+    background_tasks.add_task(background_task)
+    return {}
+
+
+uvicorn.run(app)
+```
+
+A workaround is to explicitly put the new spans in their own trace using [
+`attach_context`][logfire.propagate.attach_context]:
+
+```python
+from logfire.propagate import attach_context
+
+
+async def background_task():
+   # `attach_context({})` forgets existing context
+   # so that spans within start a new trace.
+   with attach_context({}):
+      with logfire.span('new trace'):
+         await asyncio.sleep(0.2)
+         logfire.info('background')
+
+```
+
 ## Custom head sampling
 
 If you need more control than random sampling, you can pass an [OpenTelemetry