# L'hopital's Rule.

• Oct 18th 2008, 10:47 PM
U-God
L'hopital's Rule.
I have to use l'hopital's rule to show that for any fixed t, the limit as omega approaches 1, for $\displaystyle \frac{2}{1-\omega^2} (cos(\omega t) - cos(t) ) = tsin(t)$

To do this, I assume I must differentiate with respect to omega, so that means I am allowed to take t as a constant value?

Even if I am, I can't seem to be able to get there.. Any help would be greatly appreciated, cheers.
• Oct 18th 2008, 11:06 PM
mr fantastic
Quote:

Originally Posted by U-God
I have to use l'hopital's rule to show that for any fixed t, the limit as omega approaches 1, for $\displaystyle \frac{2}{1-\omega^2} (cos(\omega t) - cos(t) ) = tsin(t)$

To do this, I assume I must differentiate with respect to omega, so that means I am allowed to take t as a constant value?

Even if I am, I can't seem to be able to get there.. Any help would be greatly appreciated, cheers.

Aha, so you have to do the DE that way.

Yes, you differentiate wrt $\displaystyle \omega$, which means you treat t as a constant:

$\displaystyle \lim_{\omega \rightarrow 1} \frac{2(\cos (\omega t) - \cos t}{1 - \omega^2} = \lim_{\omega \rightarrow 1} \frac{2 (- t \sin (\omega t) - 0)}{-2 \omega} = \lim_{\omega \rightarrow 1} \frac{-2 t \sin (\omega t)}{-2 \omega} = \, ....$
• Oct 18th 2008, 11:08 PM
U-God
Thanks again mr fantastic,
I don't know why I couldn't do that!! I haven't done limits all semester and I was trying to differentiate the function as a whole, not the top and bottom individually.

Cheers,