Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

Bereken $https://latex.codecogs.com/png.image?\large \dpi{110}\displaystyle\: \lim_{x \to e}\left ( \ln x \right ) ^{\frac{1}{x-e}}$	A. 1
B. e
C.
D.
E.

Evaluate $https://latex.codecogs.com/png.image?\large \dpi{110}\displaystyle\: \lim_{x \to e}\left ( \ln x \right ) ^{\frac{1}{x-e}}$	A. 1
	B. e
	C.
	D.
	E. ∞

$x=\log 10^x\qquad x=e^{\ln x}\\\displaystyle\:\lim_{x\to e}\left(\ln x\right)^{\frac{1}{x-e}}=\displaystyle\:\lim_{x\to e}e^{{\ln(\ln x)}^{\frac{1}{x-e}}}=\displaystyle\:\lim_{x\to e}e^{\frac{\ln(\ln x)}{x-e}}=e^{\displaystyle\:\lim_{x\to e}e^{\frac{\ln(\ln x)}{x-e}}}\\\overset{H}{=}e^{\displaystyle\:\lim_{x\to e}e^{\left(\frac{\frac1{\ln x}.\frac1x}{1}\right)}}=e^{\displaystyle\:\lim_{x\to e}e^{\left(\frac{1}{x.\ln x}\right)}}=e^{\frac{1}{e.1}}=\sqrt[e]{e}$