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

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We are given the function f(x) = (3x - 12) / (x - 4).

A function is continuous at x= c if:

- f(c) exists.
- lim x-->c f(x) exists.
- lim x-->c f(x) = f(c).

Now we see that for the function f(x) = (3x - 12) / (x - 4) if we substitute the value of x with 4, we see that f(4) = (3*4 - 12) / ( 4 - 4) = 0/0, which is an indeterminate value.Therefore f(4) does not exist. This make the function discontinuous.

**The function f(x) = (3x - 12) / (x - 4) is therefore discontinuous with the point of discontinuity being x = 4. **

f(x) = (3x-12)/(x-4). Continuity of the function at x= 4.

For a curve to be continuous at a particular point x= a, the conditions are:

(i) f(x) must have a right limit = f(a+)

(ii) f(x) should have a left limit= f(a-).

(iii) f(a-) = f(a+)

(iv) f(x) should exist at a and f(a-) = f(a) = f(a+).

In the given case, the limit of f(x) as x-->4 (3x-12)/(x-4) = 3 = f(4+) = f(4-).

But the value of f(4) = (3x-12)/(x-4) = 0/0 is indeterminate.

Therefore on the graph of f(x) at x= 4 we miss a point being undefined f(4).

Therefore f(x) is dicintinuous at x= 4.

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.**