What does "passage to the limit" means?

Question

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'..."

Answer 1

Short answer: It basically means evaluating some limit.

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"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.

Answer 2

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",

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