Simplifying a function with Dirac's Delta

Question

I'm studying by myself the book Linear System and Signals from Lathi and I have a doubt in solving this exercise:

Simplify $\frac{sin(kw)}{w}\delta(w)$

But for $w=0$ the first term is not defined. Viewing the solution manual they say to use L'Hôpital's rule. I did not understand how to use L'Hopital. In my mind this rule is used to solve limits.

Calculating the limit of $\frac{sin(kw)}{w}\delta(w)$ with $w\to 0$ is correct?

If this is correct, do I have to calculate the derivative of $\delta(t)$? What's the derivative of $\delta(t)$?

Answer 1

That is why they recommend LHopital rule, to get rid of the indeterminate form. Taking the derivative will get rid of the indeterminate form. After that it appears it is just a matter of the using the product rule

Answer 2

The singularity at $w=0$ to the function $\sin(kw)/w$ is a removable singularity. This function has a continuous extension, and L'Hôpital's rule can be used to figure out what value to give at $w=0$ to remove the singularity.