Rozwiąże ktoś te dwie granice krok po kroku? Daję obliczenia, ale są jakieś skrótowe i ja ich nie rozumiem. [latex] lim_{x o +infty} frac{sqrt{1+x}-2x}{ sqrt{1+ x^{2} } } [/latex] [latex] lim_{x o +infty} frac{sqrt{1+x^(2)}-2x}{ sqrt{1+ x^{2} } } [/

[latex]displaystyle lim_{x o infty}dfrac{sqrt{1+x}-2x}{sqrt{1+x^2}}=\\ lim_{x o infty}dfrac{sqrt{x^2left(dfrac{1}{x^2}+dfrac{1}{x} ight)}-2x}{sqrt{x^2left(dfrac{1}{x^2}+1 ight)}}=\\ lim_{x o infty}dfrac{xsqrt{dfrac{1}{x^2}+dfrac{1}{x}}-2x}{xsqrt{dfrac{1}{x^2}+1}}=\\ lim_{x o infty}dfrac{sqrt{dfrac{1}{x^2}+dfrac{1}{x}}-2}{sqrt{dfrac{1}{x^2}+1}}=\\ dfrac{sqrt{0+0}-2}{sqrt{0+1}}=dfrac{-2}{1}=-2 [/latex] ---------------------------------------------------------------------- [latex]displaystyle lim_{x o infty}dfrac{sqrt{1+x^2}-2x}{sqrt{1+x^2}}=\\ lim_{x o infty}dfrac{sqrt{x^2left(dfrac{1}{x^2}+1 ight)}-2x}{sqrt{x^2left(dfrac{1}{x^2}+1 ight)}}=\\ lim_{x o infty}dfrac{xsqrt{dfrac{1}{x^2}+1}-2x}{xsqrt{dfrac{1}{x^2}+1}}=\\ lim_{x o infty}dfrac{sqrt{dfrac{1}{x^2}+1}-2}{sqrt{dfrac{1}{x^2}+1}}=\\ dfrac{sqrt{0+1}-2}{sqrt{0+1}}=dfrac{-1}{1}=-1 [/latex]