[tex]Bonsoir ; \\\\\\ \textit{On a : } f(x) = \dfrac{ \sqrt{x+1} + \sqrt{x^2-1} - \sqrt{x^3+1} }{x^3-1} \\\\\\ = \dfrac{ \sqrt{x+1} - \sqrt{x^3+1} }{x^3-1} + \dfrac{ \sqrt{x^2-1} }{x^3-1} \\\\\\ = \dfrac{ (\sqrt{x+1} - \sqrt{x^3+1})(\sqrt{x+1} + \sqrt{x^3+1}) }{(x^3-1)( \sqrt{x+1} + \sqrt{x^3+1} )} + \dfrac{ \sqrt{x^2-1} }{x^3-1} [/tex]
[tex]= \dfrac{ (x+1)-(x^3-1) }{(x^3-1)( \sqrt{x+1} + \sqrt{x^3+1} )} + \dfrac{ \sqrt{(x-1)(x+1)} }{(x-1)(x^2+x+1)} \\\\\\ = \dfrac{ x-x^3 }{(x-1)(x^2+x+1)( \sqrt{x+1} + \sqrt{x^3+1} )} + \dfrac{ \sqrt{(x-1)(x+1)} }{(x-1)(x^2+x+1)} \\\\\\ = \dfrac{ -x(x^2-1) }{(x-1)(x^2+x+1)( \sqrt{x+1} + \sqrt{x^3+1} )} + \dfrac{ \sqrt{x-1} \sqrt{x+1}}{ \sqrt{x-1} ^2(x^2+x+1)} [/tex]
[tex]= \dfrac{ -x(x-1)(x+1) }{(x-1)(x^2+x+1)( \sqrt{x+1} + \sqrt{x^3+1} )} + \dfrac{ \sqrt{x+1}}{ \sqrt{x-1} (x^2+x+1)} \\\\\\ = \dfrac{ -x(x+1) }{(x^2+x+1)( \sqrt{x+1} + \sqrt{x^3+1} )} + \dfrac{ \sqrt{x+1}}{ \sqrt{x-1} (x^2+x+1)} [/tex]
[tex]\\\\\\ \textit{On a : } \underset{x\rightarrow 1^+}{lim} \dfrac{ -x(x+1) }{(x^2+x+1)( \sqrt{x+1} + \sqrt{x^3+1} )} = -\dfrac{1}{3 \sqrt{2} } \\\\\\ \textit{et : } \underset{x\rightarrow 1^+}{lim} \dfrac{ \sqrt{x+1}}{ \sqrt{x-1} (x^2+x+1)} = +\infty \\\\\\ \textit{donc : } \underset{x\rightarrow 1^+}{lim} f(x) = +\infty .[/tex]
