What is the chunk update radius in single player Minecraft?

As @09stephenb stated correctly, on Single Player Minecraft, only chunks in render distance are updated. This begs the question: what is the render distance for Single Player? You can adjust it in the Options > Video Options page, at the Render Distance option.

_{(source: cursecdn.com)}

The higher the number of chunks, the farther you can see. Hence, more blocks are rendered, which might lower the game's performance. For the picture's example of 12 chunks (which is the default), you can see blocks in a 192 block radius (give or take).

Note however that chunks are 16 blocks wide, 16 blocks long, and 256 blocks deep. If you are mining very far below your farm, it will continue to be calculated. What matters most is horizontal distance.

Only chunks in render distance are are updated. You can boost this by increasing your render distance but this is more resource intensive. Render distance is just what you can see.