Answer
$-24$
Work Step by Step
The general formula for the average rate of change from $x$ to $y$ can be written as: $\dfrac{f(y)-f(x)}{y-x}$. As $y\rightarrow x$, the rate of change becomes instantaneous (derivative).
Here, we have: $f(x)=-4x^2+5$
$\lim\limits_{x\to 3}\dfrac{f(x)-f(3)}{x-3}=\lim\limits_{x\to 3}\dfrac{(-4x^2+5)-(-4(3)^2+5)}{x-3} \\=\lim\limits_{x\to 3}\dfrac{-4(x^2-9)}{x-3} \\=\lim\limits_{x\to 3}\dfrac{-4(x-3)(x+3)}{x-3} \\=\lim\limits_{x\to 3} -4 (x+3) \\=-4(3+3) \\=(-4)(6) \\=-24$
