I was wondering if there is any upper limit we can impose on x, like the largest representable number. Otherwise we can always find some x that when raised to the appropriate power will be larger than the factorial in the denominator. I may be misunderstanding something here. Thanks.
You can assume that x is within representable range, but this will not be the best you can do. Please use the (easy) properties of cosine, as mentioned in the extra assignment notes I posted, to find a suitable not too large x.