determine the limit shown below in simplest form. lim x→−1 x² + x / -3x - 3

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+x=x(x + 1)$. The denominator $-3x-3=-3(x + 1)$. So the function becomes $\lim_{x\rightarrow - 1}\frac{x(x + 1)}{-3(x + 1)}$.

Step2: Simplify the function

Cancel out the common factor $(x + 1)$ (since $x\neq - 1$ when taking the limit), we get $\lim_{x\rightarrow - 1}\frac{x}{-3}$.

Step3: Evaluate the limit

Substitute $x=-1$ into $\frac{x}{-3}$, we have $\frac{-1}{-3}=\frac{1}{3}$.

Answer:

$\frac{1}{3}$