Final answer:
Among the given options, g(x) is the only function that is continuous on the interval 0 < x < 5, as others have either discontinuities or are not defined. Therefore, the correct answer is Option D: II and III only.
Step-by-step explanation:
The student's question asks which among the given functions are continuous on the interval 0 < x < 5. To determine whether each function is continuous, we must look for any points of discontinuity within this interval.
I. f(x) = (x-3) / (x2-9): This function is not continuous on 0 < x <5 because it has a discontinuity at x=3, arising from the denominator becoming zero, which results in division by zero.
II. g(x) = (x-3) / (x2+9): This function is continuous because the denominator never becomes zero, the numerator is a polynomial, and it does not have any discontinuities on the interval 0 < x < 5.
III. h(x) = ln(x-3): This function will be continuous on its domain, but since the natural logarithm is not defined for non-positive arguments, the domain of h(x) starts at x=3. Thus, h(x) is not continuous on the entire interval 0 < x < 5 as it does not exist for x in the range from 0 < x < 3.
From the given options, only g(x) is continuous for all x in the range 0 < x < 5, which makes Option D the correct choice: II and III only.