The rational reasons for denying this city existance in that time :
*first : Nazareth was not mentioned in old testament(not even once), even in Joshua: 18 and 19 , when it was counting the city and the villages of Palestine , it didn't mention any thing about Nazareth , specially these verses which described the region in details
Joshua 19:10-16
19:10 And the third lot came up for the children of Zebulun according to their families: and the border of their inheritance was unto Sarid:
19:11 And their border went up toward the sea, and Maralah, and reached to Dabbasheth, and reached to the river that is before Jokneam;
19:12 And turned from Sarid eastward toward the sunrising unto the border of Chislothtabor, and then goeth out to Daberath, and goeth up to Japhia,
19:13 And from thence passeth on along on the east to Gittahhepher, to Ittahkazin, and goeth out to Remmonmethoar to Neah;
19:14 And the border compasseth it on the north side to Hannathon: and the outgoings thereof are in the valley of Jiphthahel:
19:15 And Kattath, and Nahallal, and Shimron, and Idalah, and Bethlehem: twelve cities with their villages.
19:16 This is the inheritance of the children of Zebulun according to their families, these cities with their villages.
*second : the talmud mentioned 63 village in Galilee but Nazareth was not mentioned.
*third : Paul did not know any thing about Nazareth , he did not mentioned it, not even once .
However, he mentioned Jesus 221 times, non of them did he call him Jesus the Nazarene.
*forth : no historian or Geographical did mention the city before the 4th century.
to be continued .....