1)
![\lim_{x \to +\infty}\left[x - \sqrt[]{x^2 + 1} \right]](/latexrender/pictures/c961676c8de614d4e66658cb4d68ac6a.png "\lim_{x \to +\infty}\left[x - \sqrt[]{x^2 + 1} \right]")
![\lim_{x \to +\infty}x - \sqrt[]{x^2 + 1}. \frac{x + \sqrt[]{x^2 + 1}}{x + \sqrt[]{x^2 + 1}}](/latexrender/pictures/7ba0c5178c07359ceaedd2ab92122343.png "\lim_{x \to +\infty}x - \sqrt[]{x^2 + 1}. \frac{x + \sqrt[]{x^2 + 1}}{x + \sqrt[]{x^2 + 1}}")
![\lim_{x \to +\infty}\frac{x^2 - (x^2 + 1)}{x + \sqrt[]{x^2 + 1}}](/latexrender/pictures/91858e1639b0c969bf945e43eab571e5.png "\lim_{x \to +\infty}\frac{x^2 - (x^2 + 1)}{x + \sqrt[]{x^2 + 1}}")
![\lim_{x \to +\infty}\frac{-1}{x + \sqrt[]{x^2 + 1}}](/latexrender/pictures/23c46d0212d11d9fdb9ae2f12e49d5b7.png "\lim_{x \to +\infty}\frac{-1}{x + \sqrt[]{x^2 + 1}}")
Esta resolução é válida? Mais precisamente as operações com o símbolo "infinito"...
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2)
![\lim_{x \to +\infty}\left[\sqrt[]{x + 1} - \sqrt[]{x + 3} \right]](/latexrender/pictures/c342929165de1483956d055da0400c45.png "\lim_{x \to +\infty}\left[\sqrt[]{x + 1} - \sqrt[]{x + 3} \right]")
![\lim_{x \to +\infty}\left[\sqrt[]{x + 1} - \sqrt[]{x + 3} \right].\frac{\left[\sqrt[]{x + 1} + \sqrt[]{x + 3} \right]}{\left[\sqrt[]{x + 1} + \sqrt[]{x + 3} \right]}](/latexrender/pictures/780768895269a25f6c9065798159215c.png "\lim_{x \to +\infty}\left[\sqrt[]{x + 1} - \sqrt[]{x + 3} \right].\frac{\left[\sqrt[]{x + 1} + \sqrt[]{x + 3} \right]}{\left[\sqrt[]{x + 1} + \sqrt[]{x + 3} \right]}")
![\lim_{x \to +\infty}\frac{x + 1 - x - 3}{\sqrt[]{x+1}+\sqrt[]{x+3}}](/latexrender/pictures/2590aacedf54c7e8f029e8b8dd4d7632.png "\lim_{x \to +\infty}\frac{x + 1 - x - 3}{\sqrt[]{x+1}+\sqrt[]{x+3}}")
![\lim_{x \to +\infty}\frac{-2}{\sqrt[]{x+1}+\sqrt[]{x+3}}](/latexrender/pictures/daa42efc20f7c42503c2e33e07f61d2a.png "\lim_{x \to +\infty}\frac{-2}{\sqrt[]{x+1}+\sqrt[]{x+3}}")
![\lim_{x \to +\infty}\frac{1}{\sqrt[]{x}}.\frac{-2}{\sqrt[]{1 + \frac{1}{x}}+\sqrt[]{1+\frac{3}{x}}}](/latexrender/pictures/4b6ce970c964061ccd72fc956f3da18e.png "\lim_{x \to +\infty}\frac{1}{\sqrt[]{x}}.\frac{-2}{\sqrt[]{1 + \frac{1}{x}}+\sqrt[]{1+\frac{3}{x}}}")
Que propriedade, envolvendo os radicandos, foi usada na linha acima?
Desde já agradeço!
. No segundo, ele apenas colocou o número x em evidência, veja: 
.
![\frac{\sqrt[]{\sqrt[4]{8}+\sqrt[]{\sqrt[]{2}-1}}-\sqrt[]{\sqrt[4]{8}-\sqrt[]{\sqrt[]{2}-1}}}{\sqrt[]{\sqrt[4]{8}-\sqrt[]{\sqrt[]{2}+1}}}](/latexrender/pictures/981987c7bcdf9f8f498ca4605785636a.png "\frac{\sqrt[]{\sqrt[4]{8}+\sqrt[]{\sqrt[]{2}-1}}-\sqrt[]{\sqrt[4]{8}-\sqrt[]{\sqrt[]{2}-1}}}{\sqrt[]{\sqrt[4]{8}-\sqrt[]{\sqrt[]{2}+1}}}")
![\sqrt[]{2}](/latexrender/pictures/f21662d1cabab6e8b273a4b6f1cd663a.png "\sqrt[]{2}")
e elevar ao quadrado os dois lados)