Short answers, pulled from the story.
Grégoire de Saint-Vincent provided the first explicit definition of a limit in his 1647 work Opus Geometricum. He described the terminus of a geometric series as the end that no progression can reach even if continued to infinity. This definition marked a shift from static geometry to a dynamic understanding of approaching values.
Isaac Newton stated in the Scholium to his 1687 work that ultimate ratios are not actually ratios of ultimate quantities but limits which quantities can approach so closely that their difference is less than any given quantity. Bruce Pourciau argues that Newton provided the first epsilon argument over a century before it was formalized. This insight established that a difference can be made smaller than any arbitrary quantity.
Augustin-Louis Cauchy formalized the definition of the limit of a function in 1821 following the work of Bernard Bolzano. Karl Weierstrass also contributed to formalizing this definition which became known as the epsilon-delta definition. This definition states that for every real number epsilon there exists a delta such that for any x satisfying the condition the function value is within epsilon of the limit.
The limit of the sequence 0.999... is exactly 1. This sequence demonstrates that a value can be approached arbitrarily closely and in the limit the difference between the sequence and the target value becomes zero. The distance between 0.999... and 1 is not just small but non-existent in the limit.
The derivative is defined as the limit of the difference quotient as h approaches 0. This definition relies on the limit of the function f(x + h) - f(x) divided by h as h approaches 0. The derivative measures the instantaneous rate of change but its existence depends entirely on the limit of the difference quotient.
A Cauchy sequence of real numbers is defined such that for every real number epsilon there is an n such that whenever m and k are greater than n the distance between the mth and kth terms is less than epsilon. For sequences of real numbers any Cauchy sequence is convergent and they are equivalent. A metric space in which every Cauchy sequence is also convergent is known as a complete metric space.