The value of p, so that $\underset{x\to \infty }{\lim }{x}^p(\sqrt[3]{x+1}+\sqrt[3]{x-1}-2\sqrt[3]{x})$ exists and is equal to some finite non-zero real number, is equal to

Put,  $t=\frac{1}{x}\Rightarrow \underset{t\to 0}{\lim }{t}^{-p}({(\frac{1}{t}+1)}^{\frac{1}{3}}+{(\frac{1}{t}-1)}^{\frac{1}{3}}-2{\left(\frac{1}{t}\right)}^{\frac{1}{3}})$

$\Rightarrow \underset{t\to 0}{\lim }{t}^{-p\frac{1}{3}}({(1+t)}^{\frac{1}{3}}+{(1-t)}^{\frac{1}{3}}-2)$  use expansion 

$$\begin{gathered} {(1+t)}^{\frac{1}{3}}=1+\frac{t}{3}-\frac{t^2}{9};{(1-t)}^{\frac{1}{3}}=1-\frac{t}{3}-\frac{t^2}{9} \\ \Rightarrow \underset{t\to 0}{\lim }{t}^{-p\frac{1}{3}}(-\frac{2t^2}{9})\Rightarrow -\frac{2}{9}{t}^{-p+2\frac{1}{3}}=-\frac{2}{9}.t^{-p+\frac{5}{3}}so,p=\frac{5}{3}\Rightarrow 3\rho =5 \end{gathered}$$