The Evolution of the Idea of a Limit
The definition of limit is one that has evolved over many centuries. The following "definitions" are found in the article
Touring the Calculus Gallery
by
William Dunham
which appeared in the January 2005 issue of The American Mathematical Monthly
Newton (1642 - 1727) defined what we now call the derivative as
the ultimate ratio of vanishing quantities
where the "ultimate ratio" was characterized as
the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish
L'Hospital (1661 - 1704) defined the differential as
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The infinitely small part by which a variable quantity is continually increased or decreased is called the differential (Difference) of that quantity |
Cauchy (1789 - 1857) defined a limit as
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When the values successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all the others |