Limits, textbook error?

Why would the limit not exist? Shouldn't it be "The limit as x --> 1- is infinity" ?

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limit = ∞ implies DNE. – Don Larynx Sep 11 '13 at 19:52
Without evaluating the limit, one possible reason is that it approaches different values from the positive and negative sides of $1$. The terminology is correct (as noted in the answers below), but certain books may choose to say "the limit as $x\to 1$ increases without bound." However, as @Jean-Sébastien notes, this is a one-sided limit and the directions argument is not valid. – abiessu Sep 11 '13 at 19:52
The real concern here is the method used to "prove" the limit doesn't exist. – Rebecca J. Stones Sep 11 '13 at 19:54
@abiessu the limit is only taken from the left side of one – Jean-Sébastien Sep 11 '13 at 19:54
That method... I'd never accept that, you can build a function that goes as high as you want before making it go to zero or any other number, and that method would give an $\infty$ limit when there is a finite limit. – MyUserIsThis Sep 11 '13 at 20:07

You need to calculate $$\lim_{x \to 1, \\ x<1} \frac{\arctan x}{\arccos x}$$

You calculate separatley the limits at the numerator and denominator: $$\lim_{x \to 1, \\ x<1} \arctan x=\arctan{1}=\pi/4$$ $$\lim_{x \to 1, \\ x<1} \arccos x=0_+$$ (the plus sign stands for the fact that the limit is zero and it approaches zero on the positive side; see the graph below)

So the result is $1/0_+=+\infty$

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For a limit to 'exist' it must converge to a finite value. Else it does not make sense to talk about a limit in the sense of the epsilon-delta definition. If a limit diverges to positive or negative infinity, it 'does not exist'.

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In the textbook it says that "If infinite, state whether the one-sided limits are infinity or -infinity" (I'm not sure how to do the infinity symbol so I'm just saying infinity instead of using that.) Given this info, does it make sense that it should be infinity and is just an error? Or am I missing something still? – user94360 Sep 11 '13 at 19:55
@user94360 For some basic information about writing maths at this site see e.g. here, here, here and here. – Lord_Farin Sep 11 '13 at 19:56
@user94360 No, it's not a mistake to say that $\pm \infty$ limits do not exist. Btw, the way to get the infinity symbol is to put \infty between dollar signs. More generally, you can right click on mathematical expressions and "show math as tex commands" to learn how to write stuff. Good luck! – rschwieb Sep 11 '13 at 20:01
Again, this is not an error - the limit does not exist as there is a discontinuity at $x=1$. The left-sided limit diverges to $+\infty$ and the right-sided limit to $-\infty$. This is classified as an 'essential' or in particular here, an 'infinite' discontinuity as neither a left nor a right sided limit exists since they both diverge to their respective infinities as stated above. – S Valera Sep 11 '13 at 20:04

A limit being equal to infinity is a situation where a limit does not exist. For a limit to exist, it must converge to a finite value, and since infinity is "infinite," What you could say is that the limit of f(x) as x approaches 1 increases without bound.

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In the textbook it says that "If infinite, state whether the one-sided limits are infinity or -infinity" (I'm not sure how to do the infinity symbol so I'm just saying infinity instead of using that.) Given this info, does it make sense that it should be infinity and is just an error? Or am I missing something still? – user94360 Sep 11 '13 at 19:56
Ah, I didn't know about that part; in that case, I think the book wanted to emphasize that a limit of infinity technically DNE, but if I were you, I'd put "Lim = DNE; f(X) approaches +infinity as x approaches 1 from the left" – user79790 Sep 11 '13 at 19:59
@Lord_Farin, I'm not familiar with the sgn function, but from what I gleaned from wikipedia, that limit does not exist because the limit as x approaches 0 from the left is -1 while the limit as x approaches 0 from the right is +1. Since the limit from the left doesn't equal the limit from the right, that's why the limit DNE. That is not to say that the limit is infinity, simply that it DNE. – user79790 Sep 11 '13 at 20:02
I would suggest not being so dogmatic about a limit having to be a real number, since infinite limits are certainly used in mathematics. It's kind of like insisting that $0$ is not a natural number (true in school math, false once you get to upper level undergraduate math). Calculus texts vary on how infinite limits are classified, so what's most important for user94360 at this time is how his book deals with it and having an awareness that other books can differ. Personally, I tend to to say "finite limit" or "finite or infinite limit", as needed, so as to not risk being misunderstood. – Dave L. Renfro Sep 11 '13 at 20:57

Your derivation of $$\lim_{x\to 1^-}\frac{\arctan(x)}{\arccos(x)}=\infty$$ is fine. Some author may define the above limit to be undefined. I prefer to say it diverges to infinity and reserve the word undefined for limits such as $$\lim_{x\to \infty} \sin(x), \lim_{n\to \infty} (-1)^n\ldots$$

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