The function f(x) is discontinuous because for the root of denominator, x = 3, the function is not defined.
Since the given function is a fraction, it's denominator must be different of zero for the function to be defined..
To check the continuity of a function, we'll have to determine the lateral limits of the function and the value of the function in a specific point.
We'll prove that the function has a discontinuity point for x = 3.
We'll calculate the limit of the function, when x is approaching to 3, from the left side:
lim (3x-9) / (x-3) = (3*3 - 9)/(3 - 3) = 0/0 (x->3)
Since the result is an indetermination, we'll apply L'Hospital rule:
lim (3x-9) / (x-3) = lim (3x-9)' / (x-3)'
lim (3x-9)' / (x-3)'= lim 3/1 = 3
We'll calculate the limit of the function, when x is approaching to 3, from the right side:
We notice that the right limit is equal to the left.
We'll have to determine the value of the function for x = 3.
f(x) = 0/0 not determined.
For a function to be continuous, the values of lateral limits and the value of the function have to be equal.
Since the function is not determined for x = 3, then the given function is not continuous for x = 3.