Question

What is the solution for: lim(x->2) (2-x)/(2-∛(x+6)

The answer is supposed to be 12.

Answer 1

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

Answer 2

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

Answer 3

\[\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}\]

Answer 4

next recall the identity
\[a^3-b^3 = (a-b)(a^2+ab+b^2)\]

expand the numerator