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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).
     \

Find the limit of the difference quotient for the function f(x) = 8x^2 - 5x.

Learn how to find the derivative of the quadratic function f(x) = 8x^2 - 5x using the limit definition of a derivative. This step-by-step guide explores the process of expanding the difference quotient, simplifying, and taking the limit. Suitable for students in Grades 11-12.

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