How to calculate the following limit:
$$ \lim_{n \to \infty} \left (\sqrt[3]{(n+2)^2}-\sqrt[3]{(n-2)^2} \right)$$
Attempt: I tried to get rid of the third roots by multiplying and dividing by $\sqrt[3]{(n+2)^4}+\sqrt[3]{(n+2)^2(n-2)^2}+\sqrt[3]{(n-2)^4}$, and then I simplified the numerator. I could get to
$$ \lim_{n \to \infty} \dfrac{8n^{-\frac{1}{3}}}{\sqrt[3]{\left (\frac{n+2}{n} \right)^4}+\sqrt[3]{\frac{n^2-4}{n^4}}+\sqrt[3]{\left (\frac{n-2}{n} \right)^4}}$$
See the image below for more details on my attempt.
I am not sure if my attempt is correct and I am not sure how to proceed.
Remark: The text above clearly describes the question and the attempt. The image is no longer an essential part of question. It is kept only as a reference and, if it is necessary, it can be deleted.
