Question

Lim -2X^2-6X-4
X-(-1) X^2-1

Answer 1

$\boxed{\begin{array}{l}\underline{\rm produto~de~Stevin}\\\sf (x+a)\cdot(x+b)=x^2+(a+b)x+a\cdot b\\\underline{\rm fatorac_{\!\!,}\tilde ao~pela~diferenc_{\!\!,}a~de~dois~quadrados}\\\sf a^2-b^2=(a-b)\cdot(a+b)\end{array}}$

$\boxed{\begin{array}{l}\displaystyle\sf\lim_{ x \to -1}\dfrac{-2x^2-6x-4}{x^2-1}\\\sf-2x^2-6x-4=-2\cdot(x^2+3x+2)\\\sf -2x^2-6x-4=-2\cdot(x^2+(2+1)x+2\cdot1)\\\sf -2x^2-6x-4=-2\cdot(x+2)(x+1)\\\sf x^2-1=(x-1)\cdot(x+1)\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm Vamos~substituir~no~limite}\\\displaystyle\sf\lim_{x \to -1}\dfrac{-2\cdot(x+2)\cdot\diagup\!\!\!\!\!\!(x+\diagup\!\!\!\!\!\!1)}{(x-1)\cdot\diagup\!\!\!\!\!\!(x+\diagup\!\!\!\!\!\!1)}\\\\\displaystyle\sf-2\lim_{ x \to -1}\dfrac{x+2}{x-1}\\\\\sf =-2\cdot\dfrac{-1+2}{-1-1}=-\diagdown\!\!\!\!\!\!2\cdot\dfrac{1}{-\diagdown\!\!\!\!\!\!2}=1\end{array}}$