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I read in Stewart "single variable calculus" page 83 that the limit $$\lim_{x\to 0}{1/x^2}$$ does not exist. How precise is this statement knowing that this limit is $\infty$?. I thought saying the limit does not exist is not true where limits are $\infty$. But it is said when a function does not have a limit at all like $$\lim_{x\to \infty}{\cos x}$$.

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According to some presentations of limits, it is proper to write "$\lim_{x\to 0}\frac{1}{x^2}=\infty$." This does not commit one to the existence of an object called $\infty$. The sentence is just an abbreviation for "given any $M$, there is an $\epsilon>0$ such that for all $x$ such that $0<x<\epsilon$, we have $\frac{1}{x^2}>M$." On the other hand, some presentations of limits forbid writing "$\lim_{x\to 0}\frac{1}{x^2}=\infty$." Matter of taste, pedagogical choice. – André Nicolas Apr 3 '12 at 18:13
I would accept this as an answer !! – palio Apr 3 '12 at 18:23
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You might also be interested in this: math.stackexchange.com/questions/3203/infinite-limits. – Dejan Govc Apr 3 '12 at 18:23
I'd suggest using a question mark to signify your question. – Doug Spoonwood Apr 3 '12 at 18:24

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up vote 16 down vote accepted

According to some presentations of limits, it is proper to write "$\lim_{x→0}\frac{1}{x^2}=\infty$."

This does not commit one to the existence of an object called $\infty$. The sentence is just an abbreviation for "given any real number $M$, there is a real number $\epsilon$ (which will depend on $M$) such that for $\frac{1}{x^2}>M$ for all $x$ such that $0<x<\epsilon$." It turns out that we often wish to write sentences of this type, because they have important geometric content. So having an abbreviation is undeniably useful.

On the other hand, some presentations of limits forbid writing "$\lim_{x→0}\frac{1}{x^2}=\infty$." Matter of taste, pedagogical choice. The main reason for choosing to forbid is that careless manipulation of the symbol $\infty$ all too often leads to wrong answers.

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On can write $\lim_{x\to\infty} 1/x^2 = \infty$ and at the same time say "the limit does not exist". And also say $1/x^2$ diverges to $\infty$ as $x$ goes to $\infty$. In the same way, we want to say $\sum_{n=1}^\infty 1/n = \infty$ but also to say $\sum_{n=1}^\infty 1/n$ diverges. – GEdgar Apr 3 '12 at 21:53
@GEdgar do you mean as $x$ goes to 0? – Nick T Apr 3 '12 at 23:47
$x$ goes to $0$, yes. – GEdgar Apr 3 '12 at 23:52
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A limit

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

exists if and only if it is equal to a number. Note that $\infty$ is not a number. For example $\lim_{x\to 0} \frac{1}{x^{2}} = \infty$ so it doesn't exist.

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I think we could say that infinity is not a "specific number" or infinity is not a Real number. – Emmad Kareem Apr 3 '12 at 18:43
@Emmad: What on earth is a "specific number"? You're just making things worse! – TonyK Apr 3 '12 at 19:11
@TonyK, I mean by 'specific number' a 'finite quantity'. In my understanding infinity is not a finite quantity. How does that make 'things' worse? – Emmad Kareem Apr 3 '12 at 20:02
@Emmad: Substituting your new definition into your original comment, we obtain: "I think we could say that infinity is not a 'finite quantity'...". You see how you are just making things worse? – TonyK Apr 3 '12 at 20:16
@TonyK, I hope you can help me with this; Do you consider infinity a finite quantity? I still can't see what is wrong in my definition. Please explain a bit or point me to a reference. I want to learn. Thanks. – Emmad Kareem Apr 3 '12 at 21:00
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When a function approaches infinity, the limit technically doesn't exist by the proper definition, that demands it work out to be a number. We merely extend our notation in this particular instance. The point is that the limit may not be a number, but it is somewhat well behaved and asymptotes are usually worth note.

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yes but why in other texts we find sentences like the "limit exists and is finite" – palio Apr 3 '12 at 18:11
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@palio This is probably just to emphasize that the limit really isn't infinity or negative infinity. In general the convention is to say that a limit exists only if it is equal to a number (finite numner) – Thomas Apr 3 '12 at 18:13

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