Final answer:
Using the squeeze theorem, we determine that the limit of f(x) as x approaches 3 is 6, since the bounding functions 3x - 3 and x² - 3x + 6 both have limits of 6 as x approaches 3.
Step-by-step explanation:
To find the limit of f(x) as x approaches 3, given that 3x - 3 < f(x) < x² - 3x + 6 for x > 0, we can make use of the squeeze theorem. The squeeze theorem states that if g(x) ≤ f(x) ≤ h(x) as x approaches some value, and the limits of g(x) and h(x) as x approaches that value are the same, then the limit of f(x) must also be that value.
First, we find the limits of the bounding functions as x approaches 3:
- Limit of 3x - 3 as x approaches 3 is 3(3) - 3 = 6.
- Limit of x² - 3x + 6 as x approaches 3 is 3² - 3(3) + 6 = 6.
Since both bounds have the same limit as x approaches 3, by the squeeze theorem, the limit of f(x) as x approaches 3 is also 6.