aufgabe 1b

$$\begin{array}{rl}\underset{x\to 1}{\lim }\left(\frac{1}{\ln x}-\frac{x}{x-1}\right)& =\underset{x\to 1}{\lim }\frac{(x-1)-(x\ln x)}{(\ln x)(x-1)}\\ & =\underset{x\to 1}{\lim }\frac{x-1-x\ln x}{x\ln x-\ln x}\\ & \stackrel{\text{L’H}}{\underset{0/0}{=}}\underset{x\to 1}{\lim }\frac{1-0-\ln x-1}{\ln x+1-\frac{1}{x}}\\ & \stackrel{\text{L’H}}{\underset{0/0}{=}}\underset{x\to 1}{\lim }\frac{-\frac{1}{x}}{\frac{1}{x}+\frac{1}{x^2}}\\ & =\frac{-1}{1+1}\\ & =-\frac{1}{2}\end{array}$$

Ableitung von x ln x

$$\begin{array}{rl}(x\ln x{)}^{\prime }& =1\cdot \ln x+x\cdot \frac{1}{x}\\ & =\ln x+1\end{array}$$