Consider

.

In this case, there is no fraction in the limit. Since the limit of
ln(*x*) is negative infinity, we cannot use the
Multiplication Limit Law
to find this limit. We can convert the product
ln(*x*)*sin(*x*) into a fraction:

.

Now, we have a fraction where the limits of both the numerator and denominator are infinite. Thus, we can apply l'Hôpital's Rule:

.

Remember that the derivative of ln(*x*) is 1/*x*, and the derivative
of csc(*x*) is -csc(*x*)cot(*x*).

We can now use l'Hôpital's Rule again, as the limits of both the top and the bottom are zero, using the Product Rule to take the derivative of the numerator.

Copyright © 1996 Department of Mathematics, Oregon State University

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