The time difference between the last and first visible phases of the moon is known as the "New Moon Phase" (or Mahaq in Arabic). In between this invisible period, there is an instant of time, known as the "New Moon Conjunction" (or Waqt-tul Iqthiran in Arabic) when the sun, moon and earth come together in one line/plane.

If you are able to sight both the last waning crescent moon (from the eastern horizon before sunrise) and the first waxing crescent moon (from the western horizon after sunset), for a given location over a number of lunar months, then you can easily calculate the minimum period of Mahaq when it will be impossible to sight the new crescent moon by the human eye (without the use of any "calculations"!).

Experienced observers have stated that the duration of Mahaq is most commonly about 60 hours (2.5 days) long and in extremely rare cases it may be about 36 hours (1.5 days) long. It is not possible for it to be less than 24 hours (1day) long and hence they were not easily mislead by false sighting claims. The waning and waxing crescent moon photos below were taken from York (UK) in January 2013.

2013-01-10 YorkMoon

2013-01-12 YorkMoon

The duration of Mahaq from the above observations was 57 hrs. How many hours do you make Mahaq to be from your own observation experience?

The early Muslims were inspired by the Quran to look for the phases of the moon as stated below:
وَالْقَمَرَ‌ قَدَّرْ‌نَاهُ مَنَازِلَ حَتَّىٰ عَادَ كَالْعُرْ‌جُونِ الْقَدِيمِ

"And (as for) the moon, We have measured for it mansions (to traverse) till it returns like the old dried curved date stalk (Urjoonil Qadim)." [Quran 36:39]

The Mufassiroon (i.e. commentators of the Quran) have stated that "Urjoonil Qadim" mentioned above is the last waning crescent moon and not the first waxing crescent moon (called Hilal in Arabic) as mentioned by some translators of the Quran (2:189)!