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There are two different ways to approach this question. One of them, the short way, is to recognize that the given limit
`lim_(h->0) (f(x+h) - f(x))/h` is the definition of the derivative of the function f(x). Then, we can take the derivative of the given function using the rules of derivatives:
`(5-6x^2)' = -6(2x) = -12x`
This means the third choice is the right answer.
If, however, the derivatives are not to be used for this question, then we have to evaluate the limit. We can start by first working with the numerator of the given expression.
`f(x) = 5 - 6x^2`
Plug in x + h to find f(x+h):
`f(x+h) = 5-6(x+h)^2 = 5 - 6x^2 - 12xh - h^2`
Find the difference f(x+h) - f(x):
`f(x+h) - f(x) = (5-6x^2 - 12xh - h^2) - (5 - 6x^2)= -12xh - h^2`
Now consider the fraction under the limit and factor out h in the numerator:
`(f(x+h)-f(x))/h = (-12xh-h^2)/h =(-h(12x+h))/h = -12x - h`
Now we can find the limit of this as h approaches 0:
`lim_(h->0) (-12x-h) = -12x`
So we have shown again that the third choice is the correct answer.
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