Here's a slightly more detailed answer: since this is a one-variable expression, you can find the highest power of that variable in the expression (in this case, x³), and divide numerator and denominator by that monomial.
That makes the numerator:
3x² + 27 . . . . 3 . . . 27
----------- . or . ---- + -------
. . .x³ . . . . . . x . . . .x³
and the denominator
. . . 27
1 - -------
. . . x³
Now, as x -> ?, any fraction with a constant in the numerator but x in the denominator will approach zero. So the numerator approaches 0 + 0 = 0 and the denominator approaches 1 - 0 = 1.
That means the whole fraction is approaching 0/1, or zero.
See http://en.wikibooks.org/wiki/Calculus/Infinite_Limits for a good discussion of limits of rational polynomial functions as x goes to +/- infinity. There are three cases, depending upon whether the highest powers in the numerator and denominator are equal, or not. If the highest powers are not equal, then the limit (if it exists) depends upon which is greater. In this case the higher power is in the denominator, so the limit is zero. If it were in the numerator, the limit would not exist.
If the highest powers are equal, then the limit will be the ratio of their coefficients.