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?

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

  • 3
    $\begingroup$ We should restrict your first limit so that $x$ approaches zero from above (the positive side) if we want to get $+\infty$ as the limit. $\endgroup$ – hardmath Nov 10 '14 at 13:30
  • $\begingroup$ Well... The issue is, you can't approach something that is not defined. $\infty$ is not undefined. However, you can't divide by $0$, so the expression has no meaning (it is worse than being undefined by the way). $\endgroup$ – Martigan Nov 10 '14 at 13:30

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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.