Existence of a two-sided limit at a piecewise boundary point

For $x$ approaching $-1$ from the left, we use the rule $j(x) = x^3$ since that rule applies for $x \le -1$. As $x \to -1^-$, $x^3 \to (-1)^3 = -1$, so $\lim_{x \to -1^-} j(x) = -1$.

For $x$ approaching $-1$ from the right, we use the rule $j(x) = 2x+1$ since that rule applies for $x > -1$. As $x \to -1^+$, $2x+1 \to 2(-1)+1 = -1$, so $\lim_{x \to -1^+} j(x) = -1$. Since both one-sided limits equal $-1$, the two-sided limit exists and $\lim_{x \to -1} j(x) = -1$.