Question

use lhospital to determine the following limit (where \\(t\\) is regarded as a constant):

\\\lim_{\omega \to 3} \frac{\cos(\omega t) - \cos(3t)}{9 - \omega^2}\\

Explanation:

Verify the indeterminate form as \(\omega \to 3\)

$$ \lim_{\omega \to 3} \frac{\cos(\omega t) - \cos(3t)}{9 - \omega^2} = \frac{\cos(3t) - \cos(3t)}{9 - 9} = \frac{0}{0} $$

Apply L'Hopital's Rule by differentiating with respect to \(\omega\)

$$
LATEXBLOCK0
$$

Evaluate the limit of the quotient of derivatives

$$ \lim_{\omega \to 3} \frac{-t\sin(\omega t)}{-2\omega} = \frac{-t\sin(3t)}{-6} = \frac{t\sin(3t)}{6} $$

Answer:

Use L'Hospital to determine the following limit (where \(t\) is regarded as a constant):

$$\lim_{\omega\to 3} \frac{\cos (\omega t) - \cos (3t)}{9 - \omega^2}$$

<blank>\(\frac{t\sin(3t)}{6}\)</blank>