xlogxの極限，グラフ，積分など

$x=\dfrac{1}{y}$ とおくと，

$\displaystyle\lim_{x\to +0}x\log x\\ =\displaystyle\lim_{y\to +\infty}\dfrac{1}{y}\log \dfrac{1}{y}\\ =\displaystyle\lim_{y\to +\infty}\dfrac{-\log y}{y}=0$

$\displaystyle\lim_{x\to +0}x\log x\\ =\displaystyle\lim_{x\to +0}\dfrac{\log x}{\frac{1}{x}}\\ =\displaystyle\lim_{x\to +0}\dfrac{\frac{1}{x}}{-\frac{1}{x^2}}\\ =\displaystyle\lim_{x\to +0}(-x)\\ =0$

$(x\log x)'=x\cdot\dfrac{1}{x}+1\cdot \log x=1+\log x$