[latex] \lim_{n \to 0} 1-cos x/ \sqrt{1+x} -1[/latex]

[latex]1-cosx=2sin ^{2} \frac{x}{2} [/latex] [latex] \lim_{x \to 0} \frac{1-cosx}{ \sqrt{1+x}-1}= \lim_{x \to 0} \frac{2sin^2 \frac{x}{2}\cdot(\sqrt{1+x}+1) }{ (\sqrt{1+x}-1)( \sqrt{1+x}+1) }= \\ \\ = \lim_{x \to 0} \frac{2sin^2 \frac{x}{2}\cdot(\sqrt{1+x}+1) }{ (\sqrt{1+x})^2-1}= \lim_{x \to 0} \frac{2sin^2 \frac{x}{2}\cdot( \sqrt{1+x}+1) }{x}= \\ \\ \lim_{x \to 0} \frac{sin \frac{x}{2} }{ \frac{x}{2} }\cdot \lim_{x \to 0}sin \frac{x}{2}\cdot \lim_{x \to 0} (\sqrt{1+x}+1)= \\ \\ =1\cdot 0\cdot 2=0[/latex]