Question

$lim \: (\sqrt{x} - x ) \div x - 1 \\ \\ x -> 1$
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Answer 1

Explicação passo-a-passo:

$\sf \lim_{x\to~1}~\dfrac{\sqrt{x}-x}{x-1}$

$\sf =\lim_{x\to~1}~\dfrac{\sqrt{x}-x}{x-1}\cdot\dfrac{\sqrt{x}+x}{\sqrt{x}+x}$

$\sf =\lim_{x\to~1}~\dfrac{(\sqrt{x})^2-x^2}{(x-1)\cdot(\sqrt{x}+x)}$

$\sf =\lim_{x\to~1}~\dfrac{x-x^2}{(x-1)\cdot(\sqrt{x}+x)}$

$\sf =\lim_{x\to~1}~\dfrac{x\cdot(1-x)}{(x-1)\cdot(\sqrt{x}+x)}$

$\sf =\lim_{x\to~1}~\dfrac{(-1)\cdot x\cdot(1-x)}{(-1)\cdot(x-1)\cdot(\sqrt{x}+x)}$

$\sf =\lim_{x\to~1}~\dfrac{-x\cdot(1-x)}{(1-x)\cdot(\sqrt{x}+x)}$

$\sf =\lim_{x\to~1}~\dfrac{-x}{\sqrt{x}+x}$

$\sf =\dfrac{-1}{\sqrt{1}+1}$

$\sf =\dfrac{-1}{1+1}$

$\sf =\dfrac{-1}{2}$