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

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

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

Answer»

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

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

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

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

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

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

\(=\frac{(\sqrt{1+0^3}+\sqrt{1+0})}{(0+1)(\sqrt{1+0^2}+\sqrt{1+0})}\)

\(= \frac{2}{1.2}\)

= 1

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

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