Mathematics - Limits Question with Solution | TestHub

${\lim }_{x\to \infty }{x}^{1/3}\left\{(x+1{)}^{1/3}+(x-1{)}^{1/3}\right\}\left\{(x+1{)}^{1/3}-(x-1{)}^{1/3}\right\}$

Rationalise ${\lim }_{x\to \infty }\frac{x^{1/3}\left\{(x+1{)}^{1/3}+(x-1{)}^{1/3}\right\}2}{\left\{(x+1{)}^{2/3}+{\left(x^2-1\right)}^{1/3}+(x-1{)}^{2/3}\right\}}$ ${\lim }_{x\to \infty }\frac{2\cdot x^{2/3}\left\{{\left(1+\frac{1}{x}\right)}^{1/3}+{\left(1-\frac{1}{x}\right)}^{1/3}\right\}2}{x^{2/3}\left\{{\left(1+\frac{1}{x}\right)}^{2/3}+{\left(1-\frac{1}{x}\right)}^{1/3}+{\left(1-\frac{1}{x}\right)}^{2/3}\right\}}=\frac{2\times 2}{3}=\frac{4}{3}$