0} \frac{\sqrt{1+x}-1}{sin(\pi(x+2))} =\\\\ lim_{x->0} \frac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{sin(\pi*x+2*\pi)*(\sqrt{1+x}+1)} =\\\\ lim_{x->0} \frac{1+x-1}{sin(\pi*x)*(\sqrt{1+x}+1)} =\\\\ lim_{x->0} \frac{x}{sin(\pi*x)*(\sqrt{1+x}+1)} =\\\\ lim_{x->0} \frac{\pi*x}{sin(\pi*x)*\pi*(\sqrt{1+x}+1)} =\\\\ \frac{1}{\pi*\sqrt{1+0}+1}=\frac{1}{2*\pi}" alt="lim_{x->0} \frac{\sqrt{1+x}-1}{sin(\pi(x+2))} =\\\\ lim_{x->0} \frac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{sin(\pi*x+2*\pi)*(\sqrt{1+x}+1)} =\\\\ lim_{x->0} \frac{1+x-1}{sin(\pi*x)*(\sqrt{1+x}+1)} =\\\\ lim_{x->0} \frac{x}{sin(\pi*x)*(\sqrt{1+x}+1)} =\\\\ lim_{x->0} \frac{\pi*x}{sin(\pi*x)*\pi*(\sqrt{1+x}+1)} =\\\\ \frac{1}{\pi*\sqrt{1+0}+1}=\frac{1}{2*\pi}" align="absmiddle" class="latex-formula">