Re: tsh ultrasensitive 0.34 what does that mean
I started with a TSH of 0.41, similar to where you are. The gyno who did the tests referred me to an endocrinologist just to follow up on that and do more tests. The tests for now are fine and my doctor is taking a wait-and-see approach, since I am asymptomatic (well, I was until my heart started getting all funky...argh!!!). His original plan was to check in with me every 6 months.
If you are not having obvious symptoms of hyperthyroidism your doctor may take the same approach. You are on the cusp of what is considered a normal TSH. I would have more tests done. You can read about a lot of them on this board - someone else may be able to tell you exactly which ones you need. I would also recommend a TSI test, which is not as common as the TSH or Free T3/T4 tests. It tests for antibodies which a "normal" person should not have.
Do you know if you have nodules or an enlarged thyroid? If you don't know, a doctor needs to look for those things.