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$$\lim_{x\rightarrow 0^-} \frac{\operatorname{arccot}(x) - \frac{\pi}{2}}{x}$$

The title says everything. I already know the limit is $+\infty$, I just want to see how it can be calculated. (Please don't use L'Hôpital's rule, I haven't covered it yet at school)

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Are you familiar to Taylor expansion? –  B. S. Mar 3 '13 at 19:11
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Please, avoid double dollar signs in titles and include your question in the body of the post, not only in the title. –  Pedro Tamaroff Mar 3 '13 at 19:11
    
Never heard about it, sadly –  Tedy S. Mar 3 '13 at 19:11
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The expression is by definition the (leftsided) derivative of $\arccot(x)$ at $x=0$. By the ways, this makes the result $-1$, not $+\infty$. –  Hagen von Eitzen Mar 3 '13 at 19:14
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@HagenvonEitzen The limit of arccot$(x)$ when $x\to0$, $x\lt0$, is $-\pi/2$, not $+\pi/2$, hence there is no derivative here and the result is not $-1$. –  Did Mar 3 '13 at 19:17

2 Answers 2

up vote 1 down vote accepted

At first a plot to show the limit is not -1.

enter image description here

The limit of $\operatorname{arccot}(x)$ as $x\to 0$ and $x<0$ is $-\frac{\pi}{2}$.

To see that, we use that $\operatorname{arccot(x)}=\frac{1}{\arctan\left(\frac{1}{x}\right)}$ and hence $$\underset{x<0}{\lim_{x\to 0}} \operatorname{arccot}(x)=\lim_{x\to -\infty} \arctan(x)=-\frac{\pi}{2}$$

So your limit is the same as $$-\underset{x<0}{\lim_{x\to 0}}\frac{\pi}{x}$$ This is the same as $$\pi \cdot \underset{x>0}{\lim_{x\to 0}} \frac{1}{x}$$

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Thanks, very good explanation, I actually understood it. –  Tedy S. Mar 4 '13 at 20:00

I assume that $\mathrm arccot x = \pi/2 - \arctan x$. Try to recover the definition of derivative for the arccot (or better: the $\arctan$) function:

$$ \lim_{x\to 0^-}\frac{\mathrm {arccot}x - \pi/2}{x} = \lim_{x\to 0^-} \frac{-\arctan x}{x} = - \arctan'(0) = -1. $$

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the derivatives of arccot and arctan are bounded and hence won't get near infty anywhere –  Dominic Michaelis Mar 3 '13 at 19:43
    
We don't agree on the definition of arccot (quite a not useful function). I took the definition given in wikipedia: en.wikipedia.org/wiki/Inverse_trigonometric_functions –  Emanuele Paolini Mar 4 '13 at 6:58
    
There seems to be some variations in the definition of arccot. –  Did Mar 4 '13 at 7:20

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