Question

Salut, ma puteti ajuta la problema 520..?

Answer 1

$f(x) = x+e^x \\ \\ g\Big(f(x)\Big) = x\\ \\\\ l =\lim\limits_{x\to 0}\dfrac{\sin x}{g(e^x)} \\ \\ \\e^x =f(t)\Rightarrow e^x = t+e^t \Rightarrow x = \ln(t+e^t)\\ x \to 0 \Rightarrow t\to 0\\ \\l = \lim\limits_{t\to 0}\dfrac{\sin\Big(\ln(t+e^t)\Big)}{g\Big(f(t)\Big)} = \lim\limits_{t\to 0}\dfrac{\sin\Big(\ln(t+e^t)\Big)}{t} = \\ \\ = \lim\limits_{t\to 0}\Bigg[\dfrac{\sin\Big(\ln(t+e^t)\Big)}{\ln(t+e^t)}\cdot \dfrac{\ln(t+e^t)}{t}\Bigg] = \lim\limits_{t\to 0}\dfrac{\ln(t+e^t)}{t} =$

$= \lim\limits_{t\to 0}\dfrac{\dfrac{1+e^t}{t+e^t}}{1} = 1\\ \\\\\Rightarrow \boxed{l = 1}$