I'm not managing to calculate the following limit:
The limit as x goes to infinity of (x + cos(x)/2) / (x - sin(x)/2)
L'hopital's rule doesn't work because the limit of the quotient of the derivatives doesn't exist, and I haven't managed to find any trig identities that work. I understand intuitively that the limits is 1, since cosx and sinx are bounded and x isn't. But how can I get the function into a form where it's obvious?