Evaluate the limit.
![\[ \lim_{x \to +\infty} \left( x^2 - \sqrt{x^4 -x^2 + 1} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-25ba5d6dae4438ece3e7b1f469faf877_l3.png "Rendered by QuickLaTeX.com")
First, we multiply and divide by the conjugate of the expression, then simplify and take the limit,
![\begin{align*} \lim_{x \to +\infty}\left( x^2 - \sqrt{x^4 - x^2 + 1} \right) &= \lim_{x \to +\infty} \left( \frac{\left(x^2 - \sqrt{x^4-x^2+1} \right)\left(x^2 + \sqrt{x^4-x^2+1} \right)}{x^2 + \sqrt{x^4-x^2+1}} \right)\\[9pt] &= \lim_{x \to +\infty} \frac{x^4 - x^4 + x^2 - 1}{x^2 + \sqrt{x^4 - x^2 + 1}} \\[9pt] &= \lim_{x \to +\infty} \frac{x^2-1}{x^2 + x^2 \sqrt{1 - \frac{1}{x^2} + \frac{1}{x^4}}} \\[9pt] &= \lim_{x \to +\infty} \frac{1 - \frac{1}{x^2}}{1 + \sqrt{1-\frac{1}{x^2} + \frac{1}{x^4}}} \\[9pt] &= \frac{1}{2}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-2177e7370bbd0438abe758ba1f1551c7_l3.png "Rendered by QuickLaTeX.com")
Evaluate the limit.
First, we multiply and divide by the conjugate of the expression, then simplify and take the limit,