# Is it valid to write $\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$?

AFAIK the limes of a term does not exist if that term does not converge, but I haven't found a suiting question here yet. This probably is a double of a similar question.

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Alfe, do you mean lines, instead of limes? – yiyi Dec 9 '13 at 14:40
No …? Your question baffles me. – Alfe Dec 9 '13 at 14:40
@yiyi "limes" is Latin, in English, the word is "limit". – Daniel Fischer Dec 9 '13 at 14:40
I am going to google for what limes mean in math. THought it might be a typo, sorry. – yiyi Dec 9 '13 at 14:41
No problem, since I'm not so fluent in English mathematics, questions like these point out things to me I didn't know yet. So the case here with limes vs. limit. Thanks to @Daniel Fischer. :) – Alfe Dec 9 '13 at 14:42

Assigning a value to divergent limits is not entirely standardized among mathematicians.

One reasonable standard is that $\infty$ is not a number, while we want all limits to be numbers, so $\lim_{x\to 0}\frac{1}{x^2}$ simply doesn't exist.

Another standard, probably more common, tries to distinguish functions that grow arbitrarily large, from limits that behave erratically and don't converge. Then $\lim_{x\to 0}\frac{1}{x^2}=+\infty$. The $+$ is necessary to distinguish this case from its negative.

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That $+\infty$ thing is interesting. Does it mean that $\infty$ without the $+$ can mean positive as well as negative? In particular, can one write $\lim_{x \to 0} \frac{1}{x} = \infty$ even if that can be positive or negative? – Alfe Dec 9 '13 at 14:50
That is a reasonable standard, that as far as I know nobody uses. – vadim123 Dec 9 '13 at 14:53
@vadim, I disagree. $\infty$ means $+\infty$ by default, unless otherwise specified. Just as $99$ means $+99$. The standard way to write Alfe's assertion is $\lim_{x\to 0}|\frac1x| = \infty$. – TonyK Dec 9 '13 at 14:57
@TonyK, you've missed the point of Alfe's comment. The limit he gave is of $\frac{1}{x}$, not $|\frac{1}{x}|$, which is $+\infty$ from one side and $-\infty$ from the other. His suggested notation is $\infty$ to indicate this situation. As for your notion that $\infty=+\infty$, that's reasonable but by no means universal. – vadim123 Dec 9 '13 at 15:15
No, I haven't missed the point, I just disagree with it. You can't write $\lim_{x\to 0}\frac1x = \infty$, it's just wrong under the default interpretation. – TonyK Dec 9 '13 at 15:19

Yes, $\lim_{x\rightarrow 0 } f(x) = \infty$ means that $f(x)$ diverges to positive infinity as $x$ tends to $0$.

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Well yes, obviously $-$ but what does that mean? – TonyK Dec 9 '13 at 14:45

It is valid to write $\lim_{x \to 0} \frac{1}{x^2} = \infty$. However, this merely notation for the following: For all $M \in \mathbf{R}$ there is $\delta \in \mathbf{R}$ such that if $0<|x|<\delta$ then $\frac{1}{x^2} > M$, essentially the function $f(x)=\frac{1}{x^2}$ grows without bound as $x$ approaches $0$.

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You need to say "if $0 < |x| < \delta$". – TonyK Dec 9 '13 at 14:55

$\infty$ is a special case, but the notation is standard. The statement

$$\lim_{x\to a} f(x) = \infty$$

means that the closer $x$ approaches $a$, the larger $f(x)$ gets. In symbols:

For all $M > 0$ there exists $\delta > 0$ such that $0 < |x-a| < \delta \implies f(x)>M$.

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