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I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources.

I've read the definitions of infinitesimal given in Wikipedia and WolframMathWorld, and also related posts about the topic (like this and this) and all agree in that an infinitesimal is a quantity less than any finite quantity yet not zero (It must be in that way I guess). But what I'm not getting is the definition of infinitesimal given in my textbook:

The function $y = f(x)$ is called infinitesimal as $x \to a$ or $x \to \infty$, if $$\lim\limits_{x \to a}f(x) = 0\qquad \text{or}\qquad \lim\limits_{x \to \infty} f(x) = 0$$

  • How should I understand this?

  • Is an infinitesimal no longer thought as in the Leibnizian viewpoint but a zero limit of a function?

  • If I'm talking about infinitesimals, do I have to make clear whether I'm referring to a infinitely small quantity or to a limit zero of, say, a function? Are they the same thing? I though they don't.

From that definition:

$$\lim_{\Delta x \to 0}[f(x+\Delta x) - f(x)]$$

is and infinitesimal, and so

$$\lim_{\Delta x \to 0}\Delta x$$

Then for $\frac{\mathrm{d}y}{\mathrm{d}x}$ I have a quotient of infinitesimals.

Is the definition wrong or I'm missing something?

I've been looking for other definitions of infinitesimal relating it to the notion of limit, but haven't found one yet.

Thanks !!

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asd
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  • $\Delta x$ is the infinitesimal, i.e. it approaches 0 arbitrarily closely – Alex Feb 24 '15 at 14:56
  • It's not the limit that is an infinitesimal, it's the function itself (by your book definition of an infinitesimal) – Tryss Feb 24 '15 at 14:58
  • @Alex Ah okey. To make it clear, in the definition of the derivative \Delta x is THE infinitesimal. Does it mean that the definition of the textbook is not wrong? – asd Feb 24 '15 at 14:59
  • @Jazz: it is correct because $f(x) \to_x 0$. If it approached 5, it wouldn't be infinitesimal. – Alex Feb 24 '15 at 15:01
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    I know there are other meanings of infinitesimal, meanings in which the term infinitesimal is a noun. Using the definition in your book, the word infinitesimal is just an adjective which applied to functions and it simply abbreviates the property $\exists a\in \mathbb R\left(\lim \limits_a(f)=0\right)$. – Git Gud Feb 24 '15 at 15:03
  • @mvw Yep. Some basic properties – asd Feb 24 '15 at 15:08
  • Could you state your textbook? – mvw Feb 24 '15 at 15:28
  • Just throwing this out there: http://math.stackexchange.com/questions/822664/could-we-assign-a-numerical-value-to-an-infinitesimal –  Feb 24 '15 at 17:09

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