How to evaluate $\lim _{x\to \infty }\left(x\sqrt{x^2-x}-x^2\cos\left(\frac{1}{\sqrt{x}}\right)\right)$? I have problems to solve this limit
$$\lim _{x\to \infty }\left(x\sqrt{x^2-x}-x^2\cos\left(\frac{1}{\sqrt{x}}\right)\right)$$
I tried with Taylor:
$$\lim _{x\to \infty }\left(x^2\sqrt{1-\frac{1}{x}}-x^2\left(1-\frac{1}{2x}+\frac{1}{24x^2}\right)\right)=\lim _{x\to \infty }\left(x^2-x^2+\frac{x}{2}-\frac{1}{24}\right)=\infty$$
that is the wrong result, in must be $\color{red}{-\frac{1}{6}}$

Hint: You need to use the Taylor series expansion for the square root as well as the cosine. Expanding  $\sqrt{1-u}$ around $u=0$ we have that $$\sqrt{1-u}=1-\frac{u}{2}-\frac{u^2}{8}+O(u^3)$$ and so as $x\rightarrow \infty$ $$x^2\sqrt{1-\frac{1}{x}}=x^2-\frac{x}{2}-\frac{1}{8}+O\left(\frac{1}{x}\right).$$
