Question

Como resolver o seguinte limite: lim x tiende a 3 de (X² + 2X -15) / (raiz de (3X-6) - raiz de X?

Answer 1

$\lim_{x \to 3} \frac{x^2+2x-15}{ \sqrt{3x}-6- \sqrt{x} } \\ \\ Raizes \ do\ numerador\ x'=3 \ x"=-5 \\ \\ \lim_{x \to 3} \frac{(x-3)(x+5)}{ \sqrt{3x-6}- \sqrt{x} } \\ \\ Basta racionalizar \ :) \\ \\ \lim_{x \to 3} \frac{(x-3)(x+5)}{ \sqrt{3x-6}- \sqrt{x} }. \frac{\sqrt{3x-6}+ \sqrt{x}}{\sqrt{3x-6}+ \sqrt{x}} = \frac{(x-3)(x+5)( \sqrt{3x-6} +\sqrt{x} )}{(\sqrt{3x-6})^2- (\sqrt{x})^2} \\ \\ \lim_{x \to 3} \frac{(x-3)(x+5)( \sqrt{3x-6} +\sqrt{x} )}{3x-6- x}$
$\lim_{x \to 3} \frac{(x-3)(x+5)( \sqrt{3x-6} +\sqrt{x} )}{2x-6} = \frac{(x-3)(x+5)( \sqrt{3x-6} +\sqrt{x} )}{2(x-3)} \\ \\ \lim_{x \to 3} \frac{(x+5)( \sqrt{3x-6} +\sqrt{x} )}{2} \\ \\ Eliminamos\ a\ indeterminacao. \ :) \ Substitui \ x \\ \\ \lim_{x \to 3} \frac{(3+5)( \sqrt{3.3-6} +\sqrt{3} )}{2} \\ \\ \lim_{x \to 3} \frac{(8)( \sqrt{3} +\sqrt{3} )}{2} \\ \\ \lim_{x \to 3} \frac{(8)(2\sqrt{3})}{2} \\ \\ \lim_{x \to 3} 8\sqrt{3$