Question

Ajutor pentru limita aceasta de la UTCN!!!

Answer 1

$\displaystyle \lim\limits_{t\to 0}\int_{2^t}^{3^t}\dfrac{x}{\ln x}\, dx \overset{(*)}{=}\\ \\\\ \text{Observam ca in }x = 1\text{ nu putem calcula limita.} \\ \\ \\ w = \ln x \Rightarrow x = e^{w} \Rightarrow dx = e^{w}\, dw \\x = 2^t \Rightarrow w = t\ln 2 \\ x = 3^t \Rightarrow w = t\ln 3$

$\displaystyle \overset{(*)}{=} \lim\limits_{t\to 0}\int_{2^t}^{3^t}\dfrac{x}{\ln x}\, dx = \lim\limits_{t\to 0}\int_{t\ln 2}^{t\ln 3}\dfrac{e^{2w}}{w}\, dw = \\ \\ =\lim\limits_{t\to 0}\Big(\int_{t\ln 2}^{t\ln 3}\dfrac{e^{2w}-1}{w}\, dw +\int_{t\ln 2}^{t\ln 3}\dfrac{1}{w}\, dw\Big) \overset{(*)}{=}\\ \\\\\\ t\to 0 \Rightarrow t\ln a \to 0 \Rightarrow w\to 0 \Rightarrow \dfrac{e^{2w}-1}{w}\to 2 \Rightarrow \\ \Rightarrow \int_{t\ln 2}^{t\ln 3}\dfrac{e^{2w}-1}{w}\, dw \to 0$

$\displaystyle\overset{(*)}{=}\lim\limits_{t\to 0} \int_{t\ln 2}^{t\ln 3}\dfrac{1}{w}\, dw = \lim\limits_{t\to 0}\, \ln w\Bigg|_{t\ln 2}^{t\ln 3} =\lim\limits_{t\to 0}\ln\dfrac{t\ln 3}{t\ln 2} = \ln \dfrac{\ln 3}{\ln 2}$