Explain why the function f(x) = (3x-12) / (x-4) is discontinuous.

The function f(x) is discontinuous because for x = 4, the
function is not defined.

The given function is a ratio and a ratio
is defined when it's denominator is different from zero.

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 = 4.

We'll calculate the
left limit of the function:

lim  (3x-12) / (x-4) = (3*4 - 12)/(4 -
4) = 0/0 (x->4)

Since the result is an indetermination, we'll
apply L'Hospital rule:

lim  (3x-12) / (x-4) = lim  (3x-12)' /
(x-4)'

lim  (3x-12)' / (x-4)' = lim 3/1 =
3

Now, we'll calculate the right limit. We notice that the right
limit is equal to the left.

We'll have to determine the value of the
function for x = 4.

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.

We notice that the values of the lateral limits
are equal but the value of teh function is not determined, so the function is not continuous for
x = 4.