Soluzione Esercizio 5

 

            \begin{align} \lim_{x \to 1} \frac{x}{x-1} - \frac{1}{\ln x} & = \lim_{x \to 1} \frac{x \ln x - x + 1}{(x-1) \ln x} \\ & = \xrightarrow{l'H \hat{o} pital} \lim_{x \to 1} \frac{\ln x}{\frac{x-1}{x} + \ln x}\\ & = \lim_{x \to 1} \frac{x \ln x}{x - 1 + x \ln x} \\ & = \xrightarrow{l'H \hat{o} pital} \lim_{x \to 1} \frac{1 + \ln x}{2 + \ln x} \\ & = \frac{1}{2}, \end{align}