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![= \lim_{x \to \ 0} [ \frac{(1- \sqrt{1+x})(1+\sqrt{1+x}) }{(x^2-x)(1+\sqrt{1+x} )} ]](https://tex.z-dn.net/?f=%3D+%5Clim_%7Bx+%5Cto+%5C+0%7D+%5B+%5Cfrac%7B%281-+%5Csqrt%7B1%2Bx%7D%29%281%2B%5Csqrt%7B1%2Bx%7D%29+%7D%7B%28x%5E2-x%29%281%2B%5Csqrt%7B1%2Bx%7D+%29%7D+%5D "= \lim_{x \to \ 0} [ \frac{(1- \sqrt{1+x})(1+\sqrt{1+x}) }{(x^2-x)(1+\sqrt{1+x} )} ]")
![=\lim_{x \to \ 0} [ \frac{(1^2- (\sqrt{1+x})^2}{x(x-1)(1+\sqrt{1+x} )} ]](https://tex.z-dn.net/?f=%3D%5Clim_%7Bx+%5Cto+%5C+0%7D+%5B+%5Cfrac%7B%281%5E2-+%28%5Csqrt%7B1%2Bx%7D%29%5E2%7D%7Bx%28x-1%29%281%2B%5Csqrt%7B1%2Bx%7D+%29%7D+%5D "=\lim_{x \to \ 0} [ \frac{(1^2- (\sqrt{1+x})^2}{x(x-1)(1+\sqrt{1+x} )} ]")
![=\lim_{x \to \ 0} [ \frac{(1- (1+x)}{x(x-1)(1+\sqrt{1+x} )} ]](https://tex.z-dn.net/?f=%3D%5Clim_%7Bx+%5Cto+%5C+0%7D+%5B+%5Cfrac%7B%281-+%281%2Bx%29%7D%7Bx%28x-1%29%281%2B%5Csqrt%7B1%2Bx%7D+%29%7D+%5D "=\lim_{x \to \ 0} [ \frac{(1- (1+x)}{x(x-1)(1+\sqrt{1+x} )} ]")
![=\lim_{x \to \ 0} [ \frac{(1- 1-x)}{x(x-1)(1+\sqrt{1+x} )} ]](https://tex.z-dn.net/?f=%3D%5Clim_%7Bx+%5Cto+%5C+0%7D+%5B+%5Cfrac%7B%281-+1-x%29%7D%7Bx%28x-1%29%281%2B%5Csqrt%7B1%2Bx%7D+%29%7D+%5D "=\lim_{x \to \ 0} [ \frac{(1- 1-x)}{x(x-1)(1+\sqrt{1+x} )} ]")
![=\lim_{x \to \ 0} [ \frac{(-x)}{x(x-1)(1+\sqrt{1+x} )} ]](https://tex.z-dn.net/?f=%3D%5Clim_%7Bx+%5Cto+%5C+0%7D+%5B+%5Cfrac%7B%28-x%29%7D%7Bx%28x-1%29%281%2B%5Csqrt%7B1%2Bx%7D+%29%7D+%5D "=\lim_{x \to \ 0} [ \frac{(-x)}{x(x-1)(1+\sqrt{1+x} )} ]")
![=\lim_{x \to \ 0} [ \frac{(-1)}{(x-1)(1+\sqrt{1+x} )} ]](https://tex.z-dn.net/?f=%3D%5Clim_%7Bx+%5Cto+%5C+0%7D+%5B+%5Cfrac%7B%28-1%29%7D%7B%28x-1%29%281%2B%5Csqrt%7B1%2Bx%7D+%29%7D+%5D "=\lim_{x \to \ 0} [ \frac{(-1)}{(x-1)(1+\sqrt{1+x} )} ]")
