"The old man goes to the doctor because his vision is bad."
Translation:Den gamle mand tager til lægen fordi hans syn er svagt.
Because it refers to a man (or a part of a man) and not a thing. If the sentence was "Den gamle dame tager til lægen..." it would be "hendes" instead of "hans".
An example of "sit" would be "Manden kigger på sit ur" (The man looks at his watch). In this case it refers to the watch which is a thing and therefore "sit" is used.