Floating point: insignificant differences add up to large differences.
Mathematicians defining limits: *sweat nervously* "So um (*Autistic screeching*) well, the limit is just the sum of series, basically(*inhales*): Its the result. The Science is Settled."
The modern definition of a limit (for any ε there exists an index N so that ...) was given by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticed at the time), and by Karl Weierstrass in the 1870s.
https://en.wikipedia.org/wiki/Limit_of_a_sequence#History