The problem asks for the limit of a difference quotient, which represents the derivative of the function $f(x) = 8x^2 - 5x$. Specifically, it is asking for:

$\lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$

Steps to solve this:

1. Find $f(x + \Delta x)$:

   • Given $f(x) = 8x^2 - 5x$, substitute $x + \Delta x$ into the function.
   • $f(x + \Delta x) = 8(x + \Delta x)^2 - 5(x + \Delta x)$.

2. Expand the expression:

   • Expand $(x + \Delta x)^2$ to get $x^2 + 2x\Delta x + (\Delta x)^2$.
   • Substitute back into the function: [ f(x + \Delta x) = 8(x^2 + 2x\Delta x + (\Delta x)^2) - 5(x + \Delta x). \