Question

lim┬(x→4)⁡((-5x² + 20x)/(x³ - 3x² - 4x)) = \boxed{}

Explanation:

Step1: Factor numerator and denominator

Factor numerator: $-5x^2 + 20x = -5x(x - 4)$
Factor denominator: $x^3 - 3x^2 - 4x = x(x^2 - 3x - 4) = x(x - 4)(x + 1)$

Step2: Simplify the fraction

Cancel out common factors $(x - 4)$ and $x$ (for $x
eq 0, 4$):
$\frac{-5x(x - 4)}{x(x - 4)(x + 1)} = \frac{-5}{x + 1}$

Step3: Evaluate the limit as $x \to 4$

Substitute $x = 4$ into $\frac{-5}{x + 1}$:
$\frac{-5}{4 + 1} = \frac{-5}{5} = -1$

Answer:

$-1$