You ask if the limit is infinity (or any other value). Why not just evaluate the expression with x = 100,000,000 (a large number).
That is calculate cos(100,000,000) + 100,000,000/(7-2*100,000,000). What do you get? The limit will be close to that. Now go and try to figure out how to get the correct answer. Remember that cos x will simply fluctuate between -1 and 1 so keep that in mind when you calculate cos(100,000,000).