[Math] Roman numerals Subtractive VS Additive

notation

What is the Difference between Additive notation and Subtractive notation in Roman numerals?

correct me if im wrong but this is an example of Additive:
XXXIII = 10 + 10 + 10 + 1 + 1 + 1 = 33

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IIII and IV are both used for 4. IIII is additive; IV is subtractive.

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[Math] Limits notation: equals or arrow

I think the first one is right and the second is wrong. When the limit existis it is surely a number, and a number doesn't tend to anything.

Maybe he meant something like $$f(x)\overset{x\to \infty}{\to} 0$$ But I would prefer always $$\lim_{x\to \infty} f(x)=0$$ because the limit IS something and not tends to something.

When you use the arrows you say something like it tends to so essentially you say in $f(x)\to 0$ as $x\to \infty$ that when $x$ goes to $\infty$ your function tends to zero.

The second notation would be, as the limit is a fixed $c$, $$\text{c}\to 0$$ which I think is nonsense.

The notation to use depends on wheter you are in a text or you are in display mode. In a display mode I would use $$\lim_{x\to \infty} f(x)=0$$ In inline there are three options, the first is

$\lim_{x\to \infty} f(x)=0$

$f(x)\to 0$ as $x\to \infty$

As $x$ goes to infinity $f(x)$ tends to zero.

Personally I prefer the third, because in the first the index will be hardly legible, in the second there are to many mathematical symbols in a sentence and the third will be the easiest to read.

[Math] the difference between $x \bmod y$ and $x \pmod y$

In my experience, the expression $x\bmod y$ is a binary operation which returns the remainder when you divide $y$ into $x$.

By contrast, the parenthesized $\pmod y$ is used to qualify an equivalence. Thus one might write "$x_1\equiv x_2 \pmod y$". Usually the parenthesized $\pmod y$ goes at the right of the line, right-justified. So it is a qualifier which tells you which equivalence relation is intended. Therefore I think they have a different meaning.

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[Math] Limits notation: equals or arrow

[Math] the difference between $x \bmod y$ and $x \pmod y$

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