calculus - Help with $lim_{xrightarrow +infty} (x^2 - sqrt{x^4 - x^2 + 1})$

$\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1}) = ?$

I don't know how to solve the indetermination there... is it possible to rearrange the expression in brackets in order to use L'Hospital or Taylor Series?

Answer

Hint: $$x^2-\sqrt{x^4-x^2+1}=\left(x^2-\sqrt{x^4-x^2+1}\right)\cdot\frac{x^2+\sqrt{x^4-x^2+1}}{x^2+\sqrt{x^4-x^2+1}}=\frac{(x^2)^2-\left(\sqrt{x^4-x^2+1}\right)^2}{x^2+\sqrt{x^4-x^2+1}}$$

Once you have that simplified, multiply by $\dfrac{1/x^2}{1/x^2}$ and recognize that $x^2=\sqrt{x^4}$ to distribute the $1/x^2$ 'into' the radical.