For positive constants 
, and 
with 
, the displacement of a weight on a spring is given by the function
![\[ f(t) = \frac{A}{c^2 - k^2} (\sin (kt) - \sin (ct)). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ace96c2de83495fa1c1eb6a2c3673d55_l3.png "Rendered by QuickLaTeX.com")
Determine
![\[ \lim_{c \to k} f(t). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-34fc9ccf647841b79d8d315f277c0277_l3.png "Rendered by QuickLaTeX.com")
Using L’Hopital’s rule we compute the limit,
![\begin{align*} \lim_{c \to k} f(t) &= \lim_{c \to k} \frac{A(\sin (kt) - \sin (ct))}{c^2 - k^2} \\[9pt] &= \lim_{c \to k} \frac{-At \cos (ct)}{2c} \\[9pt] &= -\frac{At \cos (kt)}{2k}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e46fc348837e3ed1cfa738d5ca97572e_l3.png "Rendered by QuickLaTeX.com")
For positive constants , and with , the displacement of a weight on a spring is given by the function
Determine
Using L’Hopital’s rule we compute the limit,