Answer:
Explanation:
To find the intervals in which the function \( f(x) = \frac{\ln(x)}{x^2 - 9x + 20} \) is continuous, we need to consider the domain of the function. For a rational function like this one, we need to look for values of \( x \) that make the denominator equal to zero because division by zero is undefined.
The denominator \( x^2 - 9x + 20 \) factors as \( (x-4)(x-5) \). Therefore, the values of \( x \) that make the denominator zero are \( x = 4 \) and \( x = 5 \).
Now, we need to find the intervals where the function is continuous. These intervals are determined by the values of \( x \) where both the numerator and the denominator are defined. So, we need to exclude the values of \( x \) that make the denominator zero.
The function \( \ln(x) \) is defined only for positive \( x \), so we need to exclude \( x \leq 0 \). Also, the expression \( x^2 - 9x + 20 \) is defined for all real numbers.
Therefore, the function \( f(x) \) is continuous on the intervals:
1. \( (0, 4) \)
2. \( (4, 5) \)
3. \( (5, \infty) \)
These are the intervals in which the given function is continuous.