7: You should get
There is an
indeterminate product in
the denominator. Algebra does not seem to help, so the standard procedure
tells you to change this product into a ratio; this time you will be
"putting something up". However, none of the terms looks exactly appealing.
If you put the sine on the top, it will become even more nasty: Imagine
differentiating sine of
Remark: Is there any way to avoid this work? It depends. If you are lucky and the original difference on the top has limit zero, then you can apply the l'Hospital rule to the original fraction, which is somewhat easier (it does not have the reciprocals in it). How do you find out whether the numerator goes to zero? Just like what we did above, just ignore the sine.