The limit is zero because the logarithm of a large positive number is always smaller than the number itself, so the denominator is always greater than the numerator, and takes precedence when considering the limit.
(Taking logs of top and bottom: ln(ln(x))/n(ln(x)). The log-log numerator is small compared to the denominator, which is amplified by the multiplier n, showing that the limit as x approaches infinity must be zero.)