Question

how to do the limit as x approaches 0 of (sin4x)/(tan5x)

Answer 1

4/5

http://www.wolframalpha.com/input/?i=Limit%5BCot%5B5+x%5D+Sin%5B4+x%5D%2C+x+-%3E+0%5D&t=sftb01

Answer 2

If you know l'Hopital's rule, this wouldn't be a bad place to use it.

If not or if you want to avoid the careful writing out of l'Hopital rule application, then write
\[\frac{\sin 4x}{\tan 5x} = \frac{4}{5} \frac{(\sin 4x)/ 4x}{(\tan 5x)/5x}\]

Then the limit of that last fraction is 1/1