Note that 9/10, 10/11, and 11/12 are all less than one, so , , and all tend to zero as x goes to infinity. The largest of these numbers is 11/12, so it goes to zero most slowly. Thus, we multiply both the numerator and denominator by :
Note that 120/121 and 54/55 are both less than one, as expected, and so as x goes to infinity, the numerator of the above goes to zero, and the denominator goes to one. Thus the limit is 0.