Find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), where \( h \neq 0 \), for the function below. \[ f(x)=\frac{5}{x-8} \] Simplify your answer as much as possible.

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), we first need to evaluate \( f(x+h) \): \[ f(x+h) = \frac{5}{(x+h) - 8} = \frac{5}{x + h - 8} \] Next, we substitute \( f(x) \) and \( f(x+h) \) into the difference quotient: \[ \frac{f(x+h)-f(x)}{h} = \frac{\frac{5}{x+h-8} - \frac{5}{x-8}}{h} \] To combine the fractions in the numerator, we find a common denominator, which is \((x+h-8)(x-8)\): \[ \frac{5(x-8) - 5(x+h-8)}{(x+h-8)(x-8)} \] Simplifying the numerator: \[ 5(x-8) - 5(x+h-8) = 5x - 40 - (5x + 5h - 40) = -5h \] Thus, we have: \[ \frac{-5h}{(x+h-8)(x-8)} \] Now, substituting this back into the difference quotient gives us: \[ \frac{-5h}{(x+h-8)(x-8)} \cdot \frac{1}{h} = \frac{-5}{(x+h-8)(x-8)} \] Finally, we obtain the simplified form of the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{-5}{(x+h-8)(x-8)} \]