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Ur�ete

$$\lim\limits_{x \to \infty} x[\sqrt{(1+x^2)} - x]$$

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P�i v�po�tu t�to limity nelze pou��t v�tu o limit� sou�inu, nebo� $\lim\limits_{x \to \infty} \sqrt{(1+x^2)}= + \infty$ a $\lim\limits_{x \to \infty} x = + \infty$. Proto se pokus�me v�raz
$x[\sqrt{(1+x^2)} - x]$ vhodn� upravit:

$$x[\sqrt{(1+x^2)} - x]= x[\sqrt{(1+x^2)} - x] {\sqrt{(1+x^2)}+x \over \sqrt{(1+x^2)}+x} = {x(1+x^2-x^2) \over \sqrt{(1+x^2)} +x} = {1 \over \sqrt{1+{1 \over x^2}}+1}$$

Nyn� ji� snadno vypo�teme, �e podle v�ty o limit� spojit� funkce je:

$$\lim\limits_{x \to \infty} x[\sqrt(1+x^2) - x]={1 \over 2}$$

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