Question

I need help with this the correct answer is 1/12 and I need to know all the steps

Answer 1

`\begin{gathered} \lim _(h\to0)\frac{\sqrt[3]{8+h}-2}{h} \\ \text{Choosing number near h from left} \\ h=-1 \\ \frac{\sqrt[3]{8-1}-2}{-1}=0.087 \\ h=-0.5 \\ \frac{\sqrt[3]{8-0.5}-2}{-0.5}=0.0851 \\ h=-0.005 \\ \frac{\sqrt[3]{8-0.005}-2}{-0.005}=0.08335 \\ h=-0.0005 \\ \frac{\sqrt[3]{8-0.0005}-2}{-0.0005}=0.08333 \\ \\ \text{Choosing number near h from right} \\ h=1 \\ \frac{\sqrt[3]{8+1}-2}{1}=0.08 \\ h=0.5 \\ \frac{\sqrt[3]{8+0.5}-2}{0.5}=0.0816 \\ h=0.005 \\ \frac{\sqrt[3]{8+0.005}-2}{0.005}=0.0833 \\ h=0.0005 \\ \frac{\sqrt[3]{8+0.0005}-2}{0.0005}=0.08333 \\ \text{From the values found},\text{ it would say} \\ \lim _(h\to0)\frac{\sqrt[3]{8+h}-2}{h}=0.08333\approx(1)/(12) \end{gathered}`