given f(x) = {x^2 - 1/3x, for x != -1; -1, for x = -1. what is the limit? lim(x->-1) f(x)

Explanation:

Step1: Recall limit definition

We find the limit as $x\to - 1$ using the part of the function for $x\neq - 1$, since the limit is about values approaching $-1$ not equal to it.

Step2: Substitute $x = - 1$ into $x^{2}-\frac{1}{3}x$

Substitute $x=-1$ into $y = x^{2}-\frac{1}{3}x$. We get $(-1)^{2}-\frac{1}{3}\times(-1)$.

Step3: Simplify the expression

$(-1)^{2}=1$ and $\frac{1}{3}\times(-1)=-\frac{1}{3}$, so $1+\frac{1}{3}=\frac{3 + 1}{3}=\frac{4}{3}$.

Answer:

$\frac{4}{3}$