1. **State the problem:** We have a piecewise function $$h(z) = \begin{cases} 6z & \text{if } z \leq -4 \\ 1 - 9z & \text{if } z > -4 \end{cases}$$ We need to find: (a) $\lim_{z \to 7} h(z)$ (b) $\lim_{z \to -4} h(z)$ 2. **Recall the limit rules for piecewise functions:** - To find the limit at a point, check the function's value as $z$ approaches from the left and right. - If both one-sided limits are equal, the limit exists and equals that value. - If they differ, the limit does not exist (DNE). 3. **Evaluate (a) $\lim_{z \to 7} h(z)$:** - Since $7 > -4$, use the second piece: $h(z) = 1 - 9z$ - The function is continuous here, so $$\lim_{z \to 7} h(z) = 1 - 9(7) = 1 - 63 = -62$$ 4. **Evaluate (b) $\lim_{z \to -4} h(z)$:** - Left-hand limit ($z \to -4^-$): use $6z$ $$\lim_{z \to -4^-} h(z) = 6(-4) = -24$$ - Right-hand limit ($z \to -4^+$): use $1 - 9z$ $$\lim_{z \to -4^+} h(z) = 1 - 9(-4) = 1 + 36 = 37$$ - Since $-24 \neq 37$, the limit at $z = -4$ does not exist. **Final answers:** (a) $\lim_{z \to 7} h(z) = -62$ (b) $\lim_{z \to -4} h(z) = \text{DNE}$