# What is special with the word "limit" that it describes "approaches" in calculus?

I have been going through calculus class but could not figure out what does the word "limit" means here. I mean why do we use this Word not others. I found that the limit word means in Calculus, "a point or value which a sequence, function, or sum of a series can be made to approach progressively, until they are as close to it as desired". But can anybody simplify it?

• It's not really clear if you are asking what a limit is, or why we use the word "limit" for this concept. Jul 8, 2020 at 21:41
• This is a good question. After all, numbers do not move, so the idea of “approaching” goes against the basic nature of numbers, sequences, functions. For some people, it helps; for me, I think it misleads. Unfortunately, I’m feeling too lazy today to give you a good answer; perhaps others will. Jul 8, 2020 at 21:43
• If you start with 1 dollar and every day your money is cut in half, you have a sequence 1, 1/2, 1/4, 1/8, ..., and so on, so what is the limit of this sequence? Jul 8, 2020 at 21:56
• While almost-certainly not accurate to original intention of the term, my head-canon associates "limit" with the classic scenario of taking a step forward, then a half-step, then a quarter-step, etc. In this process, there's a limit (in the English sense) to how far one can ultimately travel (namely, twice the length of that first step). So, the term somewhat-naturally captures the sense that, sometimes, "unlimited" procedures yield "limited" results. ... After that, "limit" comes to apply generally to mean simply that there's a value at which an unbounded process settles in the long run.
– Blue
Jul 8, 2020 at 22:16

(Note: I have little idea about the actual historical origin of the term, so I am merely speculating/explaining how I believe you can sort of intuitively understand the name.)

I believe the basic idea of a limit comes from sequences of real numbers.

If you think of a nondecreasing sequence of real numbers, then the limit is exactly the supremum, i.e. the least upper bound, or the "upper limit" on the values, if you will.

For a nonincreasing sequece, the limit is the same as the infimum, which can also be colloquiually thought of as the "lower limit".

For a sequence which we know is monotone, but do not know whether it is increasing or decreasing, the limit captures the "correct" or "interesting" of the two bounds.

Of course, for an arbitrary (even convergent) sequence, the limit, the supremum, and the infimum are not equal to one another. As neither the supremum, nor the infimum capture any interesting information about the eventual behaviour of a sequence, they are not so useful if we do not care so much about the initial "noise". The notion of the limit (and also the related notions of $$\lim\sup$$ and $$\lim\inf$$) recapture what $$\sup$$ and $$\inf$$ do tell us in the monotone case.

The notion has since entered standard mathematical vocabulary to mean something that is "approached" in something that we think of as a process, and so we speak not only of limits of real-valued functions at infinity or at arbitrary points, but also of limits of functions into arbitrary topological spaces, or even of more abstract limits in which there is no obvious way of defining what an upper or lower bound might mean.

Limits can mean a boundary such as in the expression "city limits" and often when studying limits this geometric intuition applies. For example the hyperbola $$xy=1$$ has asymptotes given by the lines $$y=0$$ and $$x=0$$ which provide boundary edges for the hyperbola. If we consider the limits of $$y=1/x$$ as $$x$$ approaches infinity we see that the line that limits the curve is also the limit of the function. A second example that draws on this analogy would be the squeeze theorem which can loosely be stated as saying that if a curve if bounded by two other curves that intersect at a point, the curve they bound intersects at that point too. This geometry motivates the nomenclature.

I'll quote the entirety of the Earliest Known Uses of Some of the Words of Mathematics entry for Limit below, but I'll note here that Newton's discussion of "ultimate" velocities and ratios succinctly justifies the term thusly:

There is a limit which the velocity at the end of the motion may attain, but not exceed.

This suggests that the mathematical term is fundamentally intended to invoke the common meaning of "boundary" ... a separator between attainable and unattainable quantities.

This sense matches my head-canon (mentioned in a comment, expanded here) involving the classic scenario of taking a step forward, then a half-step, then a quarter-step, etc. In this process, there's a limit (in the English sense) to how far one can ultimately travel: namely, twice the length of that first step. You can attain any shorter distance, but none longer. And, while one can debate the (un)attainability of "twice the length of the first step" after infinitely many steps (whatever that's supposed to mean), the fact remains: that's the only distance subject to such debate, and we assign the term "limit" to that distinguished value.

(Importantly, the notion attainables and unattainables, and the separation thereof, is a bit nuanced. For instance, in considering the journey that is $$\lim_{x\to 3} x$$, there's nothing to keep $$x$$ from drifting higher or lower than ---or even onto--- the value of $$3$$; every value is "attainable" by the journeying $$x$$. However, formal definitions of limit convey that the process systematically and inevitably revokes "attainable" status from every value except $$3$$, so that $$3$$ "separates" the otherwise-empty set of attainables from the otherwise-universal set of unattainables. Again, one can debate the (un)attainability of $$3$$ itself at the end of the process (whatever that means), but only $$3$$'s status is debatable.)

From the Earliest Known Uses (emphasizing the particular sentence from Newton) ...

Gregory of St. Vincent (1584-1667) used terminus to mean the limit of a progression, according to Carl B. Boyer in The History of the Calculus and its Conceptual Development.

Isaac Newton wrote justifying limits in the Scholium to Section I of Book I of the Principia (Philosophiae Naturalis Principia Mathematica or The Mathematical Principles of Natural Philosophy) (first edition 1687)

Perhaps it may be objected, that there is no ultimate proportion, of evanescent qualities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate, velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical. (Translated by Andrew Motte 1729)

Katz (p. 471) comments, "A translation of Newton’s words into an algebraic statement would give a definition of limit close to, but not identical with, the modern one."

In 1821 Augustin-Louis Cauchy defined limit as follows: "If the successive values attributed to the same variables approach indefinitely a fixed value, such that they finally differ from it by as little as one wishes, this latter is called the limit of all the others." Cours d'analyse (Oeuvres II.3), p. 19. (Translation from Katz page 641) Cauchy introduced the modern ε, δ way of arguing.

This entry was contributed by John Aldrich. See also Limit and Delta and epsilon on the Earliest Use of Symbols of Calculus page.