Question

evaluate the limit, if it exists. (if an answer does not exist, enter dne.) \\( \lim_{x \to -4} \frac{x^2 - 2x - 24}{6x^2 + 23x - 4} \\)

Explanation:

Step1: Factor numerator and denominator

Factor $x^2-2x-24=(x+4)(x-6)$; Factor $6x^2+23x-4=(x+4)(6x-1)$

Step2: Cancel common factor

$\lim_{x \to -4} \frac{(x+4)(x-6)}{(x+4)(6x-1)} = \lim_{x \to -4} \frac{x-6}{6x-1}$

Step3: Substitute $x=-4$

$\frac{(-4)-6}{6(-4)-1}$

Step4: Simplify the expression

$\frac{-10}{-24-1}=\frac{-10}{-25}=\frac{2}{5}$

Answer:

$\frac{2}{5}$