Question : $\lim_{x\to\infty} x(\sqrt{(x^2+1)}-\sqrt[3]{(x^3+1)})$
My attempt : I simplified the problem to $\lim_{x\to\infty} x^2(\sqrt{1+\frac{1}{x^2}}-\sqrt[3]{1+\frac{1}{x^3}})$ after which I am unable to proceed. I also tried to get bounds to apply sandwich theorem. But that method too didn't work. I am unable to apply conjugate as 1 term is square root and the other is cube root. I couldn't come up with inequalities as well
I searched in stack exchange but couldn't get this sum. Any help would be appreciated.