\(\lim\limits_{x \to 0}(\frac{\sqrt {1+x^2}-\sqrt {1+x}}{\sqrt {1+x^3}-\sqrt {1+x}})\)
\(=\lim\limits_{x \to 0}\{\frac{\sqrt {1+x^2}-\sqrt {1+x}}{\sqrt {1+x^3}-\sqrt {1+x}}\times\)\(\frac{\sqrt {1+x^3}+\sqrt {1+x}}{\sqrt {1+x^3}+\sqrt {1+x}}\times\)\(\frac{\sqrt {1+x^2}+\sqrt {1+x}}{\sqrt {1+x^2}+\sqrt {1+x}}\}\)
\(=\lim\limits_{x \to 0}\{\frac{(1+x^2)- (1+x)}{(1+x^3)-(1+x)}\times\)\(\frac{\sqrt{1+x^3}+ \sqrt {1+x}}{\sqrt{1+x^2}+\sqrt{1+x}}\}\)
\(=\lim\limits_{x \to 0}\{\frac{x(x-1)}{(x^3-x)}\times\)\(\frac{\sqrt{1+x^3}+ \sqrt {1+x}}{\sqrt{1+x^2}+\sqrt{1+x}}\}\)
\(=\lim\limits_{x \to 0}\frac{x(x-1)(\sqrt{1+x^3}+\sqrt{1+x})}{x(x-1)x(x+1)(\sqrt{1+x^2}+\sqrt{1+x})}\)
\(=\lim\limits_{x \to 0}\frac{(\sqrt{1+x^3}+\sqrt{1+x})}{(x+1)(\sqrt{1+x^2}+\sqrt{1+x})}\)
\(=\frac{(\sqrt{1+0^3}+\sqrt{1+0})}{(0+1)(\sqrt{1+0^2}+\sqrt{1+0})}\)
\(= \frac{2}{1.2}\)
= 1
