determine the limit shown below in simplest form. lim x→−4 x² - 16 / 5x + 20

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}-16$ is a difference - of - squares and can be factored as $(x + 4)(x - 4)$. The denominator $5x+20$ can be factored as $5(x + 4)$. So, $\lim_{x\rightarrow - 4}\frac{x^{2}-16}{5x + 20}=\lim_{x\rightarrow - 4}\frac{(x + 4)(x - 4)}{5(x + 4)}$.

Step2: Cancel out the common factor

Since $x\neq - 4$ when taking the limit (we are approaching - 4, not equal to it), we can cancel out the common factor $(x + 4)$ in the numerator and denominator. We get $\lim_{x\rightarrow - 4}\frac{x - 4}{5}$.

Step3: Substitute the value of $x$

Substitute $x=-4$ into $\frac{x - 4}{5}$. We have $\frac{-4-4}{5}=\frac{-8}{5}$.

Answer:

$-\frac{8}{5}$