Home » Blog » Compute the limit of the given function Compute the limit of the given function by RoRi January 11, 2016 Evaluate the limit. First, we make the substitution . Then as so we have Related
![\[ \lim_{x \to +\infty} x^{\frac{1}{4}} \sin \left( \frac{1}{\sqrt{x}} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d92756b9c0141e65be0758dbc2e466a2_l3.png "Rendered by QuickLaTeX.com")
. Then 
as 
so we have![\begin{align*} \lim_{x \to +\infty} x^{\frac{1}{4}} \sin \left( \frac{1}{\sqrt{x}} \right) &= \lim_{t \to 0^+} \frac{\sin t}{\sqrt{t}} \\[9pt] &= \lim_{t \to 0^+} \frac{\cos t}{\frac{1}{2\sqrt{t}}} &(\text{L'Hopital's})\\[9pt] &= \lim_{t \to 0^+} 2 \sqrt{t} \cos t \\[9pt] &= 0. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6b9f3876c4c0d9f41104504090bbec06_l3.png "Rendered by QuickLaTeX.com")
. Then as so we have