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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.
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