Home » Blog » Find the limit of the given function Find the limit of the given function by RoRi December 22, 2015 Find the value of the following limit. We apply L’Hopital’s rule twice, Related
![\[ \lim_{x \to 0} \frac{\arcsin (2x) - 2 \arcsin x}{x^3}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-38c7c755847b79ef3bf26170edb0ffa9_l3.png "Rendered by QuickLaTeX.com")
![\begin{align*} \lim_{x \to 0} \frac{\arcsin (2x) - 2 \arcsin x}{x^3} &= \lim_{x \to 0} \frac{\frac{2}{\sqrt{1-4x^2}} - \frac{2}{\sqrt{1-x^2}}}{3x^2} \\[9pt] &= \lim_{x \to 0} \frac{ \frac{8x}{(1-4x^2)^{\frac{3}{2}}} - \frac{2x}{(1-x^2)^{\frac{3}{2}}}}{6x} \\[9pt] &= \lim_{x \to 0} \frac{ \frac{4}{(1-4x^2)^{\frac{3}{2}}} - \frac{1}{(1-x^2)^{\frac{3}{2}}}}{3} \\[9pt] &= 1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ebde8abbad82879b4fb70920ede6e735_l3.png "Rendered by QuickLaTeX.com")