Verify if the function f(x)=(3x-9)/(x-3) is discontinuous.

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.