How to calculate the limit $\lim_{x\to +\infty}x\sqrt{x^2+1}-x(1+x^3)^{1/3}$ which involves rational functions? 
Find $$\lim_{x\to +\infty}x\sqrt{x^2+1}-x(1+x^3)^{1/3}.$$ 

I have tried rationalizing but there is no pattern that I can observe. 
Edit:
So we forget about the $x$ that is multiplied to both the functions and try to work with the expression $(x^2+1)^{1/2}-(x^3+1)^{1/3}.$ Thus we have, $$(x^2+1)^{1/2}-(x^3+1)^{1/3}=\frac{((x^2+1)^{1/2}-(x^3+1)^{1/3})((x^2+1)^{1/2}+(x^3+1)^{1/3})}{(x^2+1)^{1/2}+(x^3+1)^{1/3}}$$
$$=\frac{x^2+1-(x^3+1)^{2/3}}{(x^2+1)^{1/2}+(x^3+1)^{1/3}}=\frac{(x^2+1)^2-(x^3+1)^{4/3}}{(x^2+1)^{3/2}+(x^2+1)(x^3+1)^{1/3}+x^3+1+(x^3+1)^{2/3}(x^2+1)}=??$$

You can do it with Taylor expansions:
$x\left(1+x^{2}\right)^{1/2}-x\left(1+x^{3}\right)^{1/3}=
x^{2}\left(1+\frac{1}{x^{2}}\right)^{1/2}-x^{2}\left(1+\frac{1}{x^{3}}\right)^{1/3}
 =x^{2}\left(1+\frac{1}{2}\cdot\frac{1}{x^{2}}+o\left(\frac{1}{x^{2}}\right)\right)-x^{2}\left(1+\frac{1}{3}\cdot\frac{1}{x^{3}}+o\left(\frac{1}{x^{3}}\right)\right)
 =\frac{1}{2}-\frac{1}{3x}+o\left(1\right)
 \to\frac{1}{2}$
