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 and we have Related
![\[ \lim_{x \to +\infty} x^4 \left( \cos \frac{1}{x} - 1 + \frac{1}{2x^2} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ad5a23f4eba67aa1225495b6b3eddfe4_l3.png "Rendered by QuickLaTeX.com")
, then 
as 
and we have![\begin{align*} \lim_{x \to +\infty} x^4 \left( \cos \frac{1}{x} - 1 + \frac{1}{2x^2} \right) &= \lim_{t \to 0^+} \left( \frac{\cos t}{t^4} - \frac{1}{t^4} + \frac{1}{2t^2} \right)\\[9pt] &= \lim_{t \to 0^+} \frac{ 2 \cos t - 2 + t^2}{2t^4} \\[9pt] &= \lim_{t \to 0^+} \frac{ 2t - 2 \sin t}{8t^3} &(\text{L'Hopital's})\\[9pt] &= \lim_{t \to 0^+} \frac{ 2 - 2 \cos t}{24t^2} &(\text{L'Hopital's})\\[9pt] &= \lim_{t \to 0^+} \frac{ \sin t}{24t} &(\text{L'Hopital's})\\[9pt] &= \lim_{t \to 0^+} \frac{\cos t}{24} &(\text{L'Hopital's})\\[9pt] &= \frac{1}{24}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-5452f467632ad3e0b90f16158bc37013_l3.png "Rendered by QuickLaTeX.com")
, then as and we have