I haven't done limits in a long time, so I'm not sure on how to go about solving the first one.
However, for f(x) = cos(4x) + sin(5x) the derivatives are not too hard. As sin and trig are periodic functions, the derivatives will alternate in a pattern.
For example, if f(x) = cosx, f'(x)=-sinx, f''(x)=-cosx, f'''=sinx, f''''=cosx.
So he derivatives would be:
f'(x) = -4sin(4x) + 5cos(5x)
f''(x) = -16cos(4x) - 5sin(5x)
f'''(x) = 64sin(4x) - 25cos(5x)
Then can you figure out the fourth one on your own?