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this might be a very simple question, so I apologize beforehand.

I am new to calculus and while I was investigating a bit more about it, I found this expression: "passage to the limit". I suspect it means the same thing as "as x approaches...", however, I cannot find its meaning anywhere on google.

I would appreciate your help.

EDIT: I first saw this when reading a book about the philosophy of calculus by Rene Guenon. A more practical context could be other questions asked in this forum, here is an example of them (although I cannot say for certain they mean the same thing): "Carrying out the passage to the limit under an integral sign" or "Results on passage of limit under integral sign"

EDIT 2: Here is the context in which Rene Guenon used this: "... It is by the 'law of continuity' that Leibnitz claims to justify the 'passage to the limit'..."

Theo Bendit
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GL RM
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    I've never heard of that kind of phrasing before in elementary calculus. Can you post more context? – Michael Rybkin Apr 08 '19 at 22:20
  • @MichaelRybkin I have edited the post giving more context. Thank you – GL RM Apr 08 '19 at 22:27
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    It's commonly used in France. It means ‘passing to the limit’, often to stipulate that equalities or (non strict) inequalities are preserved though the process of calculating limits. – Bernard Apr 08 '19 at 22:27
  • If the above comments don't solve your problem, can you please give a specific example of a statement that involves this terminology and that you don't understand. – Rob Arthan Apr 08 '19 at 22:39
  • @RobArthan I gave the context in which Rene Guenon used this in the new edit. Thanks. – GL RM Apr 08 '19 at 22:52
  • I was really asking for you to give a mathematical reference which was causing you difficulties rather than a philosophical one. Philosophers are notoriously vague in their use of mathematical language. – Rob Arthan Apr 08 '19 at 22:56
  • @RobArthan I am afraid that every example I find on the internet is far too advanced for me right now. As I said, I am new to calculus. Thanks. – GL RM Apr 08 '19 at 23:07
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    In the context continuity, I often hear the phrase "passing the limit into the function" (which may be what's being said here, somewhat obliquely). If $\lim_{x \to a} g(x) = L$, for some $L$, and $f$ is continuous at $L$, then $\lim_{x \to a} f(g(x)) = f\left(\lim_{x \to a} g(x) \right) = f(L)$. The $\lim_{x \to a}$ "passes" into the function's parentheses, which can only happen when $f$ is continuous at the relevant point. – Theo Bendit Apr 08 '19 at 23:22
  • @TheoBendit So this would mean that it is a limit of a function that also belongs to another function (as in "inside" the function)? Thanks. – GL RM Apr 08 '19 at 23:29
  • @Bernard could you please give an example. I know I am giving too much trouble. Sorry and thank you all. – GL RM Apr 08 '19 at 23:30
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    It really would be much better if you could give a specific example where the terminology is causing you a problem. – Rob Arthan Apr 09 '19 at 00:08
  • My impression is that 'passing to the limit' is an idiomatic phrase, at least in mathematical literature, which is often used to emphasize that the desired statement is obtained from taking limit to its approximations. – Sangchul Lee Apr 09 '19 at 00:23
  • From René Guénon's book: "...if [the limit] is effectively reached, this must not be within the continuous variation itself, nor as a final term in the indefinite sequence of gradus mutationis [degrees of change]. Nevertheless, it is by this 'law of continuity' that Leibnitz claims to justify the 'passage to the limit'..." It appears to be a discussion of the act of evaluating a limit, in the historical context of Leibniz's work, where limits have not been very clearly defined. I think it references whatever Leibniz intended by the corresponding Latin, but also the idea of evaluating a limit – Mark S. Apr 09 '19 at 11:38

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This is a common expression in French, often use in the context of infinitesimals. You establish a relation that holds for finite quantities and extrapolate to zero (or infinity), using the rules of limit computation.

E.g.

The slope of the tangent at a point of a curve is approximately obtained as the slope of a chord

$$s(x)\approx\frac{y(x+h)-y(x)}h,$$

and by "passage to the limit",

$$s(x)=y'(x).$$

  • In German you often see "Übergang" (which means "transition" or "passage"). The terming "passage" seems less popular (though not unheard of) in English. My bet is that the the author of the text the OP is reading is not a native English speaker and, hence, uses somewhat different diction. – zxmkn Mar 10 '22 at 14:59
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Short answer: It basically means evaluating some limit.

"Passing to the limit" means different things in different contexts. In the quote from René Guénon's "The Metaphysical Principles of the Infinitesimal Calculus" it just means something like "evaluating a limit", since the very concept was under contention during the time of Leibniz's work.

I've personally often heard the related phrase "passing to the limit" used to mean something like "taking the limit of both sides", as in "We have $f(x)<g(x)$ for all $x<c$. Passing to the limit and using the continuity of $f$ and $g$, we have $f(c)\le g(c)$." "By passage to the limit" could be substituted for "(by) passing to the limit" just fine, because of how English works.

If you've encountered "Carrying out the passage to the limit under an integral sign" and "Results on passage of limit under integral sign", I suspect the writers may not be native speakers of English, but it's clear that what is intended is "evaluating the limit", and the "under an integral sign" may mean moving a $\displaystyle{\lim_{x\to a}}$ from before the integral sign to after, as discussed at this MSE question.

Mark S.
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