Home » Blog » Compute the limit of the given function Compute the limit of the given function by RoRi December 25, 2015 Evaluate the limit. Let , then as . Making this substitution and using L’Hopital’s rule we have Related
![\[ \lim_{x \to +\infty} \frac{\sin \frac{1}{x}}{\arctan \frac{1}{x}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6a7bc84d0b3304e037657cbf52ea8919_l3.png "Rendered by QuickLaTeX.com")
, then 
as 
. Making this substitution and using L’Hopital’s rule we have![\begin{align*} \lim_{x \to +\infty} \frac{\sin \frac{1}{x}}{\arctan \frac{1}{x}} &= \lim_{t \to 0} \frac{\sin t}{\arctan t} \\[9pt] &= \lim_{t \to 0} \frac{\cos t}{\frac{1}{1+t^2}} \\[9pt] &= 1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-861713fb3ad8ca811c5e3652b5fa2bef_l3.png "Rendered by QuickLaTeX.com")
, then as . Making this substitution and using L’Hopital’s rule we have