Question

Salut, aveti o idee la aceasta limita ?

$\lim_{x \to \infty} sin( \sqrt{x+1})-sin( \sqrt{x} ) )$

Answer 1

$\displaystyle \sin(\sqrt{x+1})- \sin(\sqrt{x})=2 \sin \frac{\sqrt{x+1}-\sqrt{x}}{2} \cos \frac{\sqrt{x+1}+ \sqrt{x}}{2}.\\ \\ Observam~ca~\frac{\sqrt{x+1}-\sqrt{x}}{2}= \frac{1}{2(\sqrt{x+1}+ \sqrt{x})}. \\ \\ Deci \lim_{x \to \infty} \frac{\sqrt{x+1}-\sqrt{x}}{2} = 0.\\ \\ Atunci~\lim_{x \to \infty} \sin \frac{\sqrt{x+1}-\sqrt{x}}{2}=0.\\ \\ Intrucat~\cos \frac{\sqrt{x+1}+\sqrt{x}}{2}~variaza~in~intervalul~[-1,1],~rezulta\\ \\ ca~limita~initiala~este~0.$

Answer 2

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