Difference in treatment of Infinity and Undefined

Question

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?

$$ \lim_{x\to undefined}expression$$

Answer 1

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.

Answer 2

Your understanding may be a little off, if you ask me.

1. The limit of $\frac{1}{x^2}$ as $x$ approaches $0$ does, in the strict sense of the word, not exist. However, $\frac{1}{x^2}$ has the following property: $$\forall M\in\mathbb R\exists\delta>0: |x-0|<\delta \implies \frac{1}{x^2} > M.$$ Instead of writink $\frac{1}{x^2}$ has this property, we shorten our notation to $$"\lim_{x\to 0}\frac{1}{x^2} = \infty"$$

2. $\frac{1}{0}$, on the other hand, is completely different. By definition, $\frac ab$ is defined as "the unique solution of the equation $bx = a$", and by this definition, $\frac 10$ is not defined. Note, I am not saying $\frac10$ is undefined, I am saying it is not defined. It is not equal to some concept called undefined, it isn't equal to anything.