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?

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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.

Because a/b = x which means a = bx. For example : 6/2 = 3 which is same as saying 6 = 2*3. So when we divide 1 (or any number) by 0, it implies that 1/0 = x i.e. 1 = 0*x , which we know is not possible as anything multiplied by 0 gives you 0. Therefore we say it is undefined and not infinity.

Hint: Problems like these are quite easy to understand and simple to solve once we understand the underlying concepts behind the problem. To solve these types of problems effectively, we need to have some basic as well as advanced knowledge of calculus including limits, differentiation and continuity. To understand the difference between the given terms, we first of all need to know the definition of the terms. From the definition, we can easily figure out what is the difference between all of them.Complete step by step answer:Now we start off with the solution of the given problem by writing that, the definition of all the three terms are as follows,Undefined: This situation occurs for a function when for a particular value of ‘x’ there is no corresponding value of ‘y’. We say that the function is not defined or undefined at that particular value of ‘x’. For example, \\[y=\\dfrac{1}{x-1}\\] , here the function is not defined at \\[x=1\\] .Does not exist: This situation occurs when the limiting value at a particular point is not valid. In other words, the right hand limit and the left hand limit are not equal in this case.Infinity: This situation occurs when the limit exists and the value of the limit is equal to infinity.Note: Solving problems of these types requires a fair amount of idea of calculus such as limits, continuity and differentiability. We need to be very careful while evaluating the values of different limits in real problems. One term in the problem should not be confused with the other because it may lead to wrong answers in case of multiple choice questions. We should also remember that getting a limiting value equal to infinity is not equivalent to that of undefined or does not exist.

A weird looking X or N? Are you talking about aleph? Though it's not really infinity to the power of infinity...