Bonjour,
[tex]\displaystyle \lim_{x \to 0} \dfrac{\sqrt{x^2+1} -1}{x} \\\\= \lim_{x \to 0} \dfrac{(\sqrt{x^2+1} -1)*(\sqrt{x^2+1} +1)}{x*(\sqrt{x^2+1} +1)} \\\\= \lim_{x \to 0} \dfrac{x^2+1-1}{x*(\sqrt{x^2+1} +1)} \\\\= \lim_{x \to 0} \dfrac{x^2}{x*(\sqrt{x^2+1} +1)} \\\\= \lim_{x \to 0} \dfrac{x}{\sqrt{x^2+1} +1} \\\\=\dfrac{0}{2} \\\\=0\\[/tex]
