evaluate lim x→−2 3x² + 2x - 8 / -2x² - 9x - 10. submit an exact answer. provide your answer below:

Explanation:

Step1: Factor the numerator and denominator

The numerator $3x^{2}+2x - 8=(3x - 4)(x + 2)$. The denominator $-2x^{2}-9x - 10=-(2x^{2}+9x + 10)=-(2x + 5)(x+2)$.

Step2: Simplify the rational - function

$\lim_{x\rightarrow - 2}\frac{3x^{2}+2x - 8}{-2x^{2}-9x - 10}=\lim_{x\rightarrow - 2}\frac{(3x - 4)(x + 2)}{-(2x + 5)(x + 2)}=\lim_{x\rightarrow - 2}\frac{3x - 4}{-(2x + 5)}$ (since $x\neq - 2$ when taking the limit).

Step3: Substitute $x=-2$ into the simplified function

Substitute $x = - 2$ into $\frac{3x - 4}{-(2x + 5)}$, we get $\frac{3\times(-2)-4}{-[2\times(-2)+5]}=\frac{-6 - 4}{-(-4 + 5)}=\frac{-10}{-1}=10$.

Answer:

$10$