Hello
I have gone through your question.
I can understand your concern.
If your doctor recommend you to take Eltroxin for long time,i can deduct that you have a thyroid diagnose.
Eltroxine  is given when the thyroid does not produce enough of this hormone on its own and it is used to  treat 
hypothyroidism. It is also used to treat or prevent goiter (enlarged thyroid gland), which can be caused by hormone imbalances, 
radiation treatment, surgery, or cancer.
You didn't mention your exact thyroid diagnosis.This is important for me to understand better your situation and if you need to continue your tablets or not.
Anyway my suggestion is to consult an 
endocrinologist in your area because it is necessary to do the thryoid function tests (TSH ,fT3 ,fT4)to understand your actual thyroid function and a thyroid untrasound to see the structure of the gland.
These test will be helpful for the diagnosis and than we can know if you actually need thyroid 
hormone replacement therapy or not.
Hope i have clarified your doubts.
If you have other queries,feel free and ask.I will be happy to help.
Take care.