Argue that $f$ is not uniformly continuous Let $f(x)=x^3$. We want to show that $f$ is not uniformly continuous. 
Can someone please explain from $|f(x_n)-f(y_n)=\big|(n+\frac{1}{n})^3 - n^3)\big| = 3n+\displaystyle\frac{3}{n}$? How does $\big|(n+\frac{1}{n})^3-n^3\big| $ yield $3n+\displaystyle\frac{3}{n}$? ${}{}{}{}$

You're missing the $\frac{1}{n^3}$ term. Indeed, $$\left(n+\frac{1}{n}\right)^3-n^3=n^3+3n^2\frac{1}{n}+3n\frac{1}{n^2}+\frac{1}{n^3}-n^3=3n+3\frac{1}{n}+\frac{1}{n^3}$$
