Thread: finding limit

1. finding limit

What is the value of $\displaystyle \lim_{x \to \infty} (\frac{\pi}{2}-\tan^{-1} x)^{1/x}?$

2. $\displaystyle \displaystyle \lim_{x \to \infty}\left(\frac{\pi}{2} - \arctan{x}\right)^{\frac{1}{x}} = \lim_{x \to \infty}e^{\ln{\left[\left(\frac{\pi}{2} - \arctan{x}\right)^{\frac{1}{x}}\right]}}$

$\displaystyle \displaystyle = e^{\lim_{x \to \infty}\ln{\left[\left(\frac{\pi}{2} - \arctan{x}\right)^{\frac{1}{x}}\right]}}$

$\displaystyle \displaystyle = e^{\lim_{x \to \infty}\frac{1}{x}\ln{\left(\frac{\pi}{2} - \arctan{x}\right)}}$

$\displaystyle \displaystyle = e^{\lim_{x\to \infty}\frac{\ln{\left(\frac{\pi}{2} - \arctan{x}\right)}}{x}}$

This limit goes to $\displaystyle \displaystyle \frac{-\infty}{\infty}$ so you can use L'Hospital's Rule...

3. Hello, Sambit!

Another approach . . .

$\displaystyle \displaystyle\text{Find the value of: }\;\lim_{x \to \infty} \left(\tfrac{\pi}{2}-\tan^{-1}\!x\right)^{\frac{1}{x}}$

$\displaystyle \text{Let: }\,y \:=\:\left(\frac{\pi}{2} - \tan^{-1}\!x\right)^{\frac{1}{x}}$

$\displaystyle \text{Take logs: }\;\ln y \;=\;\ln\left(\frac{\pi}{2} - \tan^{-1}\!x\right)^{\frac{1}{x}} \;=\;\dfrac{\ln(\frac{\pi}{2} - \tan^{-1}\!x)}{x}$

$\displaystyle \displaystyle\text{Then: }\;\lim_{x\to\infty} \ln y \;=\;\lim_{x\to\infty}\frac{\ln(\frac{\pi}{2} - \tan^{-1}\!x)}{x} \;\to\;-\frac{\infty}{\infty}$

$\displaystyle \text{Apply l'Hopital: }\;\dfrac{\frac{\pi}{2} - \tan^{-1}\!x}{x} \;\Rightarrow\; \dfrac{\text{-}\frac{1}{1+x^2}}{1} \;=\;\dfrac{\text{-}1}{1+x^2}$

$\displaystyle \displaystyle \text{Hence: }\;\lim_{x\to\infty}\ln y \;=\;\lim_{x\to\infty}\frac{\text{-}1}{1+x^2} \;=\;0$

$\displaystyle \displaystyle\text{Therefore: }\;\lim_{x\to\infty}y \;=\;e^0 \;=\;1$

4. Originally Posted by Soroban
Hello, Sambit!

Another approach . . .

$\displaystyle \text{Let: }\,y \:=\:\left(\frac{\pi}{2} - \tan^{-1}\!x\right)^{\frac{1}{x}}$

$\displaystyle \text{Take logs: }\;\ln y \;=\;\ln\left(\frac{\pi}{2} - \tan^{-1}\!x\right)^{\frac{1}{x}} \;=\;\dfrac{\ln(\frac{\pi}{2} - \tan^{-1}\!x)}{x}$

$\displaystyle \displaystyle\text{Then: }\;\lim_{x\to\infty} \ln y \;=\;\lim_{x\to\infty}\frac{\ln(\frac{\pi}{2} - \tan^{-1}\!x)}{x} \;\to\;-\frac{\infty}{\infty}$

$\displaystyle \text{Apply l'Hopital: }\;\dfrac{\frac{\pi}{2} - \tan^{-1}\!x}{x} \;\Rightarrow\; \dfrac{\text{-}\frac{1}{1+x^2}}{1} \;=\;\dfrac{\text{-}1}{1+x^2}$

$\displaystyle \displaystyle \text{Hence: }\;\lim_{x\to\infty}\ln y \;=\;\lim_{x\to\infty}\frac{\text{-}1}{1+x^2} \;=\;0$

$\displaystyle \displaystyle\text{Therefore: }\;\lim_{x\to\infty}y \;=\;e^0 \;=\;1$

Actually it's the exact same approach

5. the limit is $\displaystyle \displaystyle\underset{x\to 0}{\mathop{\lim }}\,{{x}^{\tan x}}$ after substitution $\displaystyle t=\dfrac{\pi }{2}-\arctan x,$ now for $\displaystyle x\ne0$ we have $\displaystyle \displaystyle{{x}^{\tan x}}=\exp \left( \frac{\sin x}{\cos x}\cdot \ln x \right)=\exp \left( \frac{\sin x}{x}\cdot x\ln (x)\cdot \frac{1}{\cos x} \right),$ as $\displaystyle x\to0^+$ the limit is $\displaystyle e^0=1.$

6. Thanks to all...