I don't think that would be useful to do.
For this type of problem there is a technique that is useful.
Let y = (tan(4x))x
Then ln y = x ln(tan(4x)) = tan(4x)/(1/x)
Now take the limit of both sides, and recognize that what you're getting is the limit of the ln of what you want.
Check your textbook. I'm betting that there is an example that uses this technique.