Is the best way to approximate e by using Limits or by a infinite sum with factorials?

Assuming that you are looking for a numerical value,write out the series   for e^x

and then put  x=1 .The series converges very quickly so you don't need to take many terms to get an accurate value. 

that is e^x = 1 + x + (x^2)/2!  + (x^3)/3!   etc etc  ....  and just plug in  x=1

If you use this expansion, it will give you 50% more digits:

(1 + 1/10^100)^(10^100.5)