Question

Ma puteti ajuta , va rog, la subpunctul c).
Daca ajuta , la a) mi-a dat limita 1 si la b) am demonstrat ca functia este injectiva si surjectiva de unde a rezultat ca e bijectiva deci e inversabila

Answer 1

$f:\mathbb{R}\to \mathbb{R},\quad f(x) = x^3+x \\ \\ f^{-1}\Big(f(x)\Big) = x\Rightarrow f^{-1}(x^3+x) = x \\ \\ l =\lim\limits_{x\to \infty}\dfrac{f^{-1}(x)}{\sqrt[3]{x}} \\ \\ x = f(t) \Rightarrow x = t^3+t\\ x\to \infty \Rightarrow t^3+t \to \infty \\ \\ l = \lim\limits_{t\to \infty}\dfrac{f^{-1}(t^3+t)}{\sqrt[3]{t^3+t}} =\\ \\= \lim\limits_{t\to \infty}\dfrac{t}{\sqrt[3]{t^3+t}}=\lim\limits_{t\to \infty}\dfrac{t}{\sqrt[3]{t^3(1+\frac{1}{t})}} =$

$= \lim\limits_{t\to \infty}\dfrac{t}{t\sqrt[3]{1+\frac{1}{t}}} = \lim\limits_{t\to \infty}\dfrac{1}{\sqrt[3]{1+\frac{1}{t}}} = \\\\= \dfrac{1}{\sqrt[3]{1+0}} = \boxed{1}$