## anonymous one year ago What is the solution for: lim(x->2) (2-x)/(2-∛(x+6) The answer is supposed to be 12.

1. ganeshie8

You may use the limit definition of derivative, what is the derivative of $$\sqrt[3]{x+6}$$ at $$x=2$$ ?

2. anonymous

@ganeshie8 we haven't started with derivatives yet so idk anything about them.

3. ganeshie8

\large \begin{align} \lim\limits_{x\to 2} \dfrac{2-x}{2-\sqrt[3]{x+6}}&=\lim\limits_{x\to 2} \dfrac{x-2}{\sqrt[3]{x+6}-2}\\~\\ &=\lim\limits_{x\to 2} \dfrac{(x+6)-8}{\sqrt[3]{x+6}-\sqrt[3]{8}}\\~\\ &=\lim\limits_{x\to 2} \dfrac{\left(\sqrt[3]{x+6}\right)^3-\left(\sqrt[3]{8}\right)^3}{\sqrt[3]{x+6}-\sqrt[3]{8}}\\~\\ \end{align}

4. ganeshie8

next recall the identity $a^3-b^3 = (a-b)(a^2+ab+b^2)$ expand the numerator