Final answer:
The value that q(x)=(3x²+5x-12)/(x³-3x²) approaches as x goes to infinity is 0, as the highest power term in the denominator grows faster than the highest power term in the numerator.
Step-by-step explanation:
To determine the value that q(x)=(3x²+5x-12)/(x³-3x²) approaches as x goes to infinity, we can compare the highest powers of x in the numerator and the denominator. In this case, the highest power of x in the numerator is x² and in the denominator, it is x³. When x approaches infinity, the terms with the lower powers of x become insignificant relative to the highest power terms. Therefore, the behavior of q(x) is dominated by the ratio of the coefficients of these highest power terms. In our function, that ratio is (3/x) since the coefficient of x² is 3 and the coefficient of x³ is 1. As x goes to infinity, (3/x) approaches 0. Hence, the value that q(x) approaches as x goes to infinity is 0.