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Using limits to find the displacement of a spring

For positive constants A, c, and k with c \neq k, the displacement of a weight on a spring is given by the function

    \[ f(t) = \frac{A}{c^2 - k^2} (\sin (kt) - \sin (ct)). \]

Determine

    \[ \lim_{c \to k} f(t). \]


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*}

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