aufgabe 1c

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

4.17 (3) L'Hospital Potenzen

$$\begin{array}{rl}\underset{x\to x_0}{\lim }f(x{)}^{g(x)}& =\underset{x\to x_0}{\lim }\exp (g(x)\cdot \ln f(x))\\ & =\exp (\underset{x\to x_0}{\lim }g(x)\cdot \ln f(x))\end{array}$$

Note

$$x=\exp (\ln x)$$

Note

$$\exp (a{)}^b=\exp (a\cdot b)$$

Note

$$\lim \exp (f(x))=\exp (\lim f(x))$$