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 Calculus Calculus Math Forum

 December 7th, 2016, 11:03 PM #1 Newbie   Joined: Dec 2016 From: Việt Nam Posts: 2 Thanks: 0 Calculation limit I'm trying to figure out how to find the limit of (ln x / ln(x+1) )^xlnx , as x approaches infinity . The result is 1/e .How do I approach this? P/s : I'm not good eng, so sorry if have anything bad  December 8th, 2016, 12:03 AM #2 Member   Joined: Apr 2015 From: USA Posts: 46 Thanks: 32 You can use the approximation that $\ln(1+x)\approx x$ when $x$ approaches zero. To figure out the limit, first take the log. As a placeholder for the limit, let \begin{align}y&=\left[\frac{\ln x}{\ln(x+1)}\right]^{x\,\ln x}\cr \ln y&=x\,\ln x\cdot\ln\left[\frac{\ln x}{\ln(x+1)}\right]\cr &=x\,\ln x\cdot\ln\left[\frac{\ln x}{\ln\left(x\{1+\frac{1}{x}\}\right)}\right]\cr &=x\,\ln x\cdot\ln\left[\frac{\ln x}{\ln x+\ln\{1+\color{red}{\frac{1}{x}}\}}\right]\cr \end{align} Since $x\to\infty$ and $\color{red}{\frac{1}{x}}$ approaches zero, we can use the approximation above. \begin{align}\ln y&\approx x\,\ln x\cdot\ln\left[\frac{\ln x}{\ln x+\color{red}{\frac{1}{x}}}\right]\cr &\approx x\,\ln x\cdot\ln\left[\frac{\ln x+\frac{1}{x}}{\ln x+\frac{1}{x}}+\frac{-\frac{1}{x}}{\ln x+\frac{1}{x}}\right]\cr &\approx x\,\ln x\cdot\ln\left[1+\frac{-\frac{1}{x}}{\ln x+\frac{1}{x}}\right]\cr \end{align} The $\frac{-\frac{1}{x}}{\ln x+\frac{1}{x}}$ also approaches zero. Use the approximation above, again. \begin{align}\ln y&\approx x\,\ln x\cdot\left[\frac{-\frac{1}{x}}{\ln x+\frac{1}{x}}\right]\qquad\text{The addition of }\frac{1}{x}\text{ is negligible compared to }\ln x\text{, so}\cr &\approx x\,\ln x\cdot\left[\frac{-\frac{1}{x}}{\ln x}\right]\cr &\approx x\,\cancel{\ln x}\cdot\left[\frac{-\frac{1}{x}}{\cancel{\ln x}}\right]\cr &\approx x\cdot\left[-\frac{1}{x}\right]\cr &\approx -1\cr y&\approx e^{-1}\cr \end{align} Thanks from SenatorArmstrong December 8th, 2016, 04:21 AM #3 Newbie   Joined: Dec 2016 From: Việt Nam Posts: 2 Thanks: 0 That's great. thank you very much  Tags calculation, limit Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post evol_w10lv Calculus 2 April 25th, 2016 02:56 PM icemanfan Real Analysis 3 September 10th, 2012 12:02 PM icemanfan Calculus 2 March 9th, 2012 07:30 PM Ockonal Calculus 2 December 7th, 2011 07:04 PM Crouch Calculus 2 December 30th, 2010 02:39 AM

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