Q:

Difference in treatment of Infinity and Undefined

I understand that 
$$1)\; \lim_{x\to0}\frac1{x} = +\infty$$
$$2)\; \frac1{0} is\,undefined $$
If both infinity and undefined are just abstract concepts and not a type of number, why I see this kind of expressions used
$$ \lim_{x\to\infty}expression$$
but not these kind of expressions?
$$ \lim_{x\to undefined}expression$$

A:

While $\infty$ in calculus is not a real number, it is still a well-defined notion. It just means "eventually larger than any other point".
In the first limit you present, $\frac1x$, as $x\to 0$, we say that the limit equals $\infty$, since given any number $M$, we can find $x$ such that $\frac1x>M$. Do note, by the way, that we need $x>0$ in order to conclude the limit is positive.
In the second term you use, $\frac10$, there is no limit. We don't evaluate it against some process which continues. It's just a term which may or may not be a real number. If it were, suppose some $r$ then $r\cdot 0=1$ but there is no such $r$ which is a real number and $r\cdot 0\neq 0$.
Now, when you write $\lim\limits_{x\to y}$, you mean that either $y$ is a real number and $x$ gets arbitrarily close to it; or that $y=\pm\infty$ and $x$ gets arbitrarily large (with the appropriate sign, of course). 
So while $\infty$ is not a real number, it is a well-defined notion; whereas $\frac10$ is just meaningless.

