Question

Cum se rezolva limita asta ?

Answer 1

$\displaystyle l =\lim\limits_{x\to 2}\Big(\sqrt[3]{x-1}\Big)^{\frac{1}{x-2}} = \lim\limits_{x\to 2}\big(x-1\big)^{\frac{1}{3}\cdot \frac{1}{x-2}} =\\ \\ = \lim\limits_{x\to 2}\big[1+(x-2)\big]^{\frac{1}{x-2}\cdot \frac{1}{3}}\\ \\\\ \text{Fac schimbarea de variabila:}\\ x - 2 = t \Rightarrow t\to 0\\ \\ \\l= \Big[\lim\limits_{t\to 0}(1+t)^{\frac{1}{t}}\Big]^{\frac{1}{3}} = e^{\frac{1}{3}} = \boxed{\sqrt[3]{e}}$