TITLE: Is $\tan(\pi/2)$ undefined or infinity?
QUESTION [27 upvotes]: The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself.
If we also observe a fraction $F=a/b$ where $a,b$ are positive real numbers,  the value of $F$ increases with the decrements of $b$. Being $0$ the least non-negative integer, if $b$ tends $0$ then $F$ tends to $\infty$ which is greater than all finite numbers.  
I have also heard that no number is equal to infinity, a variable can tend to infinity. Any ratio is also a variable or is resolved to a number.
Now, 
$$\tan(\pi/2) = \frac{\sin(\pi/2)}{\cos(\pi/2)}=\frac{1}{0},$$  
is it undefined or infinity?
Also we know 
$$\tan2x= \frac{2\tan x}{1-\tan^2x}$$
Is this formula valid for $x=\pi/2$?
Any rectification is more than welcome.
Some references:

tan(PI/2) = ?
Infinity is not a number
Woah! I thought tan(pi/2) was undefined?

REPLY [4 votes]: It depends on what you mean.
Is $\tan$ a real-valued (partial) function? Then $\tan(\pi/2)$ is undefiend.
Is $\tan$ an extended real-valued (partial) function? Then $\tan(\pi/2)$ is undefined.
Is $\tan$ a projective real-valued (partial) function? Then $\tan(\pi/2) = \infty$, and $\tan$ is actually a total function on the reals. (total meaning its domain is all of the reals)