Question

Lim de (1 / x√(1+x) - 1/x) quando x tende a 0?

Answer 1

$\lim_{x \to 0} \frac{1}{x \sqrt{1+x} }- \frac{1}{x} \\\\ \lim_{x \to 0} \frac{1}{x \sqrt{1+x} }- \frac{1 \sqrt{1+x} }{x \sqrt{1+x} } \\\\ \boxed{\boxed{ \lim_{x \to 0} \frac{1- \sqrt{1+x} }{x \sqrt{1+x} } }}$

multiplica e divide por  1+√(1+x)  para ter uma diferença dos quadrados no numerador

$\lim_{x \to 0} \frac{1- \sqrt{1+x} }{x \sqrt{1+x} } * \frac{1+\sqrt{1+x}}{1+\sqrt{1+x}} \\\\ \lim_{x \to 0} \frac{1^2-( \sqrt{1+x})^2 }{(x \sqrt{1+x})*(1+ \sqrt{1+x}) } \\\\ \lim_{x \to 0} \frac{1-(1+x) }{(x \sqrt{1+x})*(1+ \sqrt{1+x}) } \\\\ \lim_{x \to 0} \frac{-\not x}{(\not x \sqrt{1+x})(1+ \sqrt{1+x}) } \\\\ \lim_{x \to 0} \frac{-1}{ (\sqrt{1+x})(1+ \sqrt{1+x}) } = \frac{-1}{( \sqrt{1+0})(1+ \sqrt{1+0}) } = \frac{-1}{2}$