limh→0(f(−2+h)−f(−2)h) where f(x)=−3x2−4x−9

To find the limit of the given expression, we first need to understand that this is the definition of the derivative of the function f(x) at x = -2. The derivative of a function at a certain point gives us the slope of the tangent line to the function at that point.

Here are the steps to find the derivative of the function f(x) = -3x^2 - 4x - 9:

1. Find the derivative of f(x) using the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).

   The derivative of -3x^2 is -6x (using the power rule), and the derivative of -4x is -4 (since the derivative of x is 1). The derivative of -9 is 0 (since the derivative of a constant is 0).

   So, the derivative of f(x) is f'(x) = -6x - 4.

2. Substitute x = -2 into the derivative to find the slope of the tangent line at x = -2.

   f'(-2) = -6*(-2) - 4 = 12 - 4 = 8.

So, the limit of the given expression is 8.