1. **Problem:** Find the one-sided and two-sided limits of $f(x)$ at given points using the graph. 2. **Recall:** The one-sided limits are $\lim_{x \to a^-} f(x)$ (from the left) and $\lim_{x \to a^+} f(x)$ (from the right). The two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal. 3. **Evaluate each:** - a) At $x = -4$: - $\lim_{x \to -4^-} f(x) = 3$ (from graph) - $\lim_{x \to -4^+} f(x) = 3$ - So, $\lim_{x \to -4} f(x) = 3$ - b) At $x = -2$: - $\lim_{x \to -2^-} f(x) = -1$ - $\lim_{x \to -2^+} f(x) = -1$ - So, $\lim_{x \to -2} f(x) = -1$ - c) At $x = 0$: - $\lim_{x \to 0^-} f(x) = 0$ - $\lim_{x \to 0^+} f(x) = 0$ - So, $\lim_{x \to 0} f(x) = 0$ - d) At $x = 2$ (vertical asymptote): - $\lim_{x \to 2^-} f(x) = -\infty$ - $\lim_{x \to 2^+} f(x) = \infty$ - Two-sided limit does not exist. - e) As $x \to -\infty$, $f(x) \to \infty$ (graph rises steeply) - f) As $x \to \infty$, $f(x) \to -2$ (horizontal asymptote) --- **Final answers:** - a) $\lim_{x \to -4} f(x) = 3$ - b) $\lim_{x \to -2} f(x) = -1$ - c) $\lim_{x \to 0} f(x) = 0$ - d) $\lim_{x \to 2} f(x)$ does not exist - e) $\lim_{x \to -\infty} f(x) = \infty$ - f) $\lim_{x \to \infty} f(x) = -2$