Question

Calculați $\lim_{x \to \infty} (\sqrt{x} -lnx)$

Answer 1

Salut,

Limita din enunț are cazul de nederminare ∞ -- ∞.

Rezolvarea se regăsește mai jos:

$\displaystyle \lim_{x \to +\infty}(\sqrt x-lnx)=\lim_{x \to +\infty}\sqrt x\cdot\left(1-\dfrac{lnx}{\sqrt x}\right)=\lim_{x \to +\infty}\sqrt x\cdot \lim_{x \to +\infty}\left(1-\dfrac{lnx}{\sqrt x}\right)=\\\\=+\infty+1-\lim_{x\to+\infty}\dfrac{lnx}{\sqrt x}=+\infty-\lim_{x\to+\infty}\dfrac{(lnx)\ '}{(\sqrt x)\ '}=+\infty-\lim_{x\to+\infty}\dfrac{\dfrac{1}x}{\dfrac{1}{2\sqrt x}}=\\\\\\=+\infty-\lim_{x\to+\infty}\dfrac{2\sqrt x}{x}=+\infty-\lim_{x\to+\infty}\dfrac{2}{\sqrt x}=+\infty-\dfrac{2}{+\infty}=+\infty-0=+\infty.$

Ai înțeles rezolvarea ?

Green eyes.