[tex]\displaystyle Avem~ \sin x \in [-1,1] ~\forall~x \in \mathbb{R} \Rightarrow 2- \sin x \in [1,3]~\forall~x \in \mathbb{R} \Rightarrow \\ \\ \Rightarrow \frac{1}{2-\sin x} \in \left[ \frac{1}{3},1 \right]. \\ \\ Ne~intereseaza~doar~faptul~ca~ \frac{1}{2- \sin x} \ge \frac{1}{3}. \\ \\ Pentru~x>0~vom~avea~ \int\limits_0^x \frac{1}{2- \sin t} \mathrm{d}t \ge \int\limits_0^x \frac{1}{3} \mathrm{d}x= \frac{x}{3}. \\ \\ Deci~F(x) \ge \frac{x}{3}~\forall~x>0.[/tex]
[tex]\displaystyle Cum~\lim_{x \to \infty} \frac{x}{3}= \infty,~rezulta~\lim_{x \to \infty}F(x)= \infty. \\ \\----------- \\ \\ Referitor~la~solutia~ta:~Nu~e~corecta,~dar~poate~fi~reglata. \\ \\ Cand~aplici~teorema~de~medie,~acel~"c"~este~in~(0,x).~Nu~este~neaparat \\ \\ acelasi~c~pentru~orice~x,~si~de~aceea~limita~initiala~nu~este~neaparat \\ \\ aceeasi~cu~limita~in~care~apare~x.~Limita~la~care~ai~ajuns~este \\ \\ de~fapt~\lim_{x \to \infty} \frac{x}{2- \sin c(x)},~ori~noi~nu~cunoastem~comportamentul[/tex]
[tex]\displaystyle functiei~c.~Stim~doar~ca~c(x) \in (0,x),~dar~asta~nu~ne~ajuta. \\ \\ Totusi,~indiferent~de~functia~c,~vom~avea~ca~in~solutia~mea \\ \\ \frac{1}{2-\sin c(x)} \ge \frac{1}{3}. \\ \\ Exista~insa~situatii~in~care~daca~nu~tinem~cont~ca~acel~"c"~este~de~fapt \\ \\ un~"c(x)",~obtinem~un~rezultat~gresit. [/tex]
