Find the limit algebraically.limh→0(f(7+h)−f(7)h) where f(x)=−8x2+7x−9

To find the limit algebraically, we first need to substitute the function f(x) into the limit expression:

lim(h→0) [(f(7+h) - f(7))/h]

Given that f(x) = -8x^2 + 7x - 9, we can substitute x = 7+h and x = 7 into the function:

f(7+h) = -8(7+h)^2 + 7(7+h) - 9 f(7) = -8(7)^2 + 7*7 - 9

Subtracting f(7) from f(7+h) gives us:

f(7+h) - f(7) = -8(7+h)^2 + 7(7+h) - 9 - (-8(7)^2 + 7*7 - 9)

Simplifying the above expression and then dividing by h gives us:

[-8(7+h)^2 + 7(7+h) - 9 + 8(7)^2 - 7*7 + 9]/h

Now, we can take the limit as h approaches 0. The terms involving h will go to zero, and we are left with the derivative of the function f(x) at x = 7, which is:

f'(x) = -16x + 7

Substituting x = 7 into the derivative gives us:

f'(7) = -16*7 + 7 = -112 + 7 = -105

So, lim(h→0) [(f(7+h) - f(7))/h] = -105.