Show why the function f(x) = (3x-12) / (x-4) is discontinuous.
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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.
You need to test if conditions of one variable continuous functions are satisfied, such that:
`lim_(x->4)(f(x) - f(4))/(x - 4) = f(4)`
Evaluating the limit yields:
`lim_(x->4)((3x - 12)/(x - 4) - f(4))/(x - 4) = f(4)`
The value `x = 4` is not in domain of definition of function because it cancels denominator `x - 4` , hence,` f(4)` is not valid.
Hence, testing the validity of statement` lim_(x->4)(f(x) - f(4))/(x - 4) = f(4)` yields that it does not hold and the function is discontinuous at `x = 4.`
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