Why a number/infinitesimal is equal to infinity? Why a number/infinitesimal+ is equal to infinity?
Why a number/infinitesimal- is equal to -infinity?
If a real number is 5, and a infinitesimal positive is 0+, is 0.1 for example, no? I got this doubt while I was doing a limit: $$\lim_{x \to -
2^-}\frac{(x-5)}{(x-2)(x+2)}$$
$$\lim_{x \to -2^-}\frac{(-2-5)}{(-2-2)(x+2)}$$
$$\lim_{x \to -2^-}\frac{-7}{(x-2)0^+}$$
$$\lim_{x \to -2^-}\frac{-7}{((-4)(0^+)}$$
$$\lim_{x \to -2^-}\frac{(-7)}{-0^+}$$
Why is equal to infinity??
 A: From the comments below, it seems the point of confusion is how to understand $0^+$. The answer is simply:
Don't use it (yet).
"$0^+$" is a shorthand used to abbreviate limit calculations, but it shouldn't be used until those precise calculations are already well-understood. In particular, rather than think about "${-7\over -0^+}$" you should force yourself to think explicitly in terms of $$\lim_{h\rightarrow 0^+}{-7\over -h}.$$

*

*Wait, isn't there a "$0^+$" in that expression already? Well, not really: it's not "the limit as $h$ approaches the number $0^+$" but rather "the limit as $h$ approaches the number $0$ from the right." Actually way back when I was learning calculus I was quite annoyed by this notation, and argued that it should be "$h\rightarrow^+0$" instead since "${}^+$" is modifying the way the limit is taken, not the value it's taken at. But anyways.

Now plugging in numbers does not a valid calculation make, but it may nonetheless be a useful reality check. Take for example $h={1\over 100000000}$. Then $${-7\over -h}={-7\over-({1\over 100000000})}={7\over({1\over 100000000})}=700000000.$$ This should suggest that if $h$ is really small and positive then ${-7\over -h}$ is really big and positive, so we should expect $\lim_{h\rightarrow 0^+}{-7\over -h}=+\infty$.
(Note that we could have simplified things right from the start and cancelled the minus signs: it's probably easier to think about $\lim_{h\rightarrow 0^+}{7\over h}$, and that's the same thing.)

In general, you need to force yourself to go slowly with limits and use the definitions carefully until you've mastered them. There are lots of places where going too fast will get you in trouble. Another common stumbling block (I've seen it frequently with students) is about the converses of the limit laws. For example, I've seen many students argue that $\lim_{x\rightarrow a}[f(x)+g(x)]$ is undefined whenever both $\lim_{x\rightarrow a}f(x)$ and $\lim_{x\rightarrow a}g(x)$ are undefined, but of course that's incorrect: consider $$f(x)={1\over x},\quad g(x)=-{1\over x}\quad\implies \quad f(x)+g(x)={1\over x}-{1\over x}.$$
