I am aware of at least three equivalent definitions of $\limsup$ and $\liminf$. I shall only write the definitions for $\limsup$. Here $(a_n)_{n \in \mathbf{N}}$ refers to a sequence of real numbers.

 1. $\limsup (a_n) := \lim_{n \to \infty} \left( \sup_{k \geq n} (a_{k}) \right)$
 2. $\limsup (a_n) := \sup \{x : x < a_n \ \text{for infinitely many} \ n \}$
 3. $\limsup (a_n) := \sup \{x : x \  \text{is a limit point of} \  (a_n)_{n \in \mathbf{N}} \}$


The concept of $\limsup$ ($\liminf$) is used all the time in mathematics, but when did mathematicians first make use of the concept? For example, did Cauchy systematically use the notion in his work? Which of the above equivalent definitions came first, and under what circumstances? I'm aware that definition $(2)$  is useful in proving the Bolzano-Weierstrass's Theorem, but I'm not sure if Bolzano and/or Weierstrass actually "called out explicitly" these concepts in their work.

Other motivating historical examples which brought $\limsup$ and $\liminf$ to the fore naturally would also help.

By the way, I'm also curious about the use of $\liminf$ in formulating the twin prime conjecture:

$\liminf_{n \to \infty} \ (p_{n+1}-p_{n}) = 2$

It's a really neat way to formulate the problem. Who first wrote it this way?