Infinity symbol

According to a Wikipedia article on the subject, in Herbert B. Enderton's book Computability: An Introduction to Recursion Theory (2011), even if nowhere else (no other reference is given, and I've never seen the usage):

If $f$ is a partial function on $S$ and $a$ is an element of $S$, then this is written as $f(a)\!\downarrow$ and is read as "$f(a)$ is defined."

If $a$ is not in the domain of $f$, then this is written as $f(a)\!\uparrow$ and is read as "$f(a)$ is undefined".

This was the most confusing doubt I had initially which my teachers could not explain satisfactorily. If you define $$\tan x=\dfrac{\sin x}{\cos x}$$ then $$\tan 90^\circ=\dfrac{\sin 90^\circ}{\cos 90^\circ}=\dfrac{1}{0}$$ which is meaningless, nonsense or undefined (whichever you want to say) as division by $0$ is not allowed. So $\tan 90^\circ$ is undefined. However, as mentioned by @Sassatelli Giulio, the following is correct to write: $$\lim_{x\to\frac{\pi}{2}^+}\tan x =-\infty$$ $$\lim_{x\to\frac{\pi}{2}^-}\tan x =\infty$$ Both of these are different things. Note that $$\lim_{x\to a}f(x)=\infty$$ is just a way of saying this: $$\forall M >0 \,\exists\, \delta>0 :\forall x \left( 0<|x-a|<\delta\implies f(x)>M\right)$$ Or in easier words: the limit grows without a bound (unbounded). And most importantly, $$\boxed{\dfrac{1}{0}\ne \infty}$$

First of all, infinity is not a real number so actually dividing something by zero is undefined. In calculus $\infty$ is an informal notion of something "larger than any finite number", but it's not a well-defined number.

You might want to read the following:

1. Is infinity a number?
2. What exactly is infinity?

Edit: the following refers to an earlier version of the question.

Secondly, as the comments remarked, $-\infty\neq+\infty$ when talking about the real line. Note that when $x<1$ we have that $x-1<0$, and when $x>1$ we have $x-1>0$. Therefore: $$\lim_{x\to1^-}\frac1{x-1}=-\infty\\\lim_{x\to1^+}\frac1{x-1}=+\infty$$

The limit is defined if and only if the two sided limits are equal. They are not, so the limit is undefined.