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I`m trying to evaluate this limit and I need some advice how to do that. $$\lim\limits_{x\to \infty}\left(\frac{2\arctan(x)}{\pi}\right)^x $$ I have a feeling it has to do with a solution of form $1^\infty$ but do not know how to proceed. Any hints/solutions/links will be appreciated

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I assume you mean $x\to+\infty$, since otherwise the limit does not exist ($\lim_{x\to-\infty}\arctan x = -\pi/2$ and $[-1]^{-\infty}$ has no sense). –  AndreasT Jun 6 '13 at 14:11

2 Answers 2

up vote 5 down vote accepted

It could be suitably modified into something involving the limit $(1+\frac1x)^x\rightarrow e$ for $x\to\infty$. $$ \left(\frac{2\arctan x}{\pi}\right)^x ~=~ \left[1 + \left(\frac{2\arctan x}{\pi}-1\right)\right]^x $$ Let $f(x)=\left(\frac{2\arctan x}{\pi}-1\right)$; clearly $f(x)\to 0$ for $x\to+\infty$, therefore $$ \left[1+f(x)\right]^{\frac{1}{f(x)}}\longrightarrow e $$ Let us focus on the limit of $xf(x)$: using l'Hospital's rule we get $$ \lim_{x\to+\infty}x\,f(x) ~=~ \lim_{x\to+\infty} \frac{\frac{2\arctan x-1}{\pi}}{\frac1x} ~\stackrel H=~ \lim_{x\to+\infty} \frac{\frac{2}{\pi(1+x^2)}}{-\frac1{x^2}} ~=~ -\frac2\pi $$ Now, putting all together: $$ \lim_{x\to+\infty} \left(\frac{2\arctan x}{\pi}\right)^x ~=~ \lim_{x\to+\infty} \big(1+f(x)\big)^x ~=~ \lim_{x\to+\infty} \left[\big(1+f(x)\big)^{\frac{1}{f(x)}}\right]^{xf(x)} ~=~ e^{-2/\pi} $$ Generally, when you run into $1^\infty$ you can work it out in this way.

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Thank you for the solution. The addition of the general method for solving such problems is welcomed. –  SteelSoul Jun 6 '13 at 14:27

Let $L$ be the limit, if it exists. Take logs of both sides:

$$\log{L} = \lim_{x \to \infty} x \log{\left ( \frac{2}{\pi} \arctan{x}\right)}$$

Note that

$$\arctan{x} \sim \frac{\pi}{2} - \frac{1}{x} \quad (x \to \infty)$$

so we have

$$\log{L} = \lim_{x \to \infty} x \log{\left ( 1-\frac{2}{\pi x}\right)} = -\frac{2}{\pi}$$

Therefore, the limit is

$$L = e^{-2/\pi}$$

ADDENDUM

I was told that the behavior of arctan I posted in the second step is not trivial. Allow me to illustrate:

$$\arctan{x} = \int_0^x \frac{dt}{t^2+1} = \int_0^{\infty} \frac{dt}{t^2+1} - \int_x^{\infty} \frac{dt}{t^2+1}$$

The first integral is $\pi/2$. The second we can approximate because we want to consider $x$ being large, which implies that $t$ is large compared to $1$. Therefore, $t^2+1 \approx t^2$ and the second integral becomes approximately

$$\int_x^{\infty} \frac{dt}{t^2} = \frac{1}{x}$$

which is what we wanted to show.

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This is a nice solution. The second step is not that trivial though. Hard to instinctively noticed that arctanx∼π/2 − 1/x (x→∞) –  SteelSoul Jun 6 '13 at 14:25
    
@SteelSoul: it's not that hard. I'll derive it for you. –  Ron Gordon Jun 6 '13 at 14:26
    
I think it follows from the fact that the cotangent is the reciprocal of the tangent. –  Bill Kleinhans Jun 6 '13 at 20:49

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