# Need help with this question: find the limit of the expression given in the image below.

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### 1 Answer

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.**