L'Hopitals rule really doesn't help in this situation. How would you determine this? As x approaches infinity, the limit of cosine is indefinite. Also as x approaches infinity, e^x approaches infinity.
So basically, are you supposed to look at it from the perspective that no matter what is going on with cosine, e^x becomes so massive that the numerator is negligible?....or how should you approach this limit?
We know that oscillates between -1 and 1. So as we still have oscillating between these values.
As . This means that the denominator gets HUGE. Divide any number by something very big, you get something very small.
So I would think that the limit tends to 0, as the denominator getting huge makes the quotient very small.