Evaluation of $\lim_{x\to -2} \frac{\sqrt[3]{x - 6} + 2}{x+2}$

Question

Evaluate $$\lim_{x\to -2} \frac{\sqrt[3]{x - 6} + 2}{x+2}$$

I tried to multiply by the conjugate expression but it didn't work. Wolfram Mathematica sais that it's a "DirectedInfinity". I don't understand that. Please, explain me the steps of solution !

Answer 1

The conjugate works for the square-root because $(a+x)(a-x)=a^2-x^2$
For the cube-root, you need to use $(a+x)(a^2-ax+x^2)=a^3+x^3$ instead.