1. **Problem Statement:** Determine the one-sided and two-sided limits of the function $f(x)$ at specified points using the graph description. 2. **Recall:** - The one-sided limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left. - The one-sided limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right. - The two-sided limit $\lim_{x \to a} f(x)$ exists only if both one-sided limits exist and are equal. 3. **Evaluate each limit:** **a) $\lim_{x \to -4} f(x)$** - From the left ($x \to -4^-$), $f(x) = 4$ (constant). - From the right ($x \to -4^+$), $f(x)$ drops steeply below 4, approaching approximately $-8$ near $x=-2$. - At $x=-4$, the function has an open circle at $( -4,4 )$. - So, $\lim_{x \to -4^-} f(x) = 4$, $\lim_{x \to -4^+} f(x)$ is much less (around $-8$), so the two-sided limit does not exist. **b) $\lim_{x \to -2} f(x)$** - From the left ($x \to -2^-$), $f(x)$ is near $-8$. - From the right ($x \to -2^+$), $f(x)$ rapidly increases towards 0 near $x=0$. - Since the function is continuous and increasing through $x=-2$, both one-sided limits approach the same value near $-8$. - So, $\lim_{x \to -2} f(x) = -8$. **c) $\lim_{x \to 0} f(x)$** - From the left ($x \to 0^-$), $f(x)$ approaches 0 (horizontal tangent at $x=-1$). - From the right ($x \to 0^+$), $f(x)$ also approaches 0. - So, $\lim_{x \to 0} f(x) = 0$. **d) $\lim_{x \to 2} f(x)$** - There is a vertical asymptote at $x=2$. - From the left ($x \to 2^-$), $f(x)$ rises sharply to $+\infty$. - From the right ($x \to 2^+$), $f(x)$ approaches $-2$ from above. - Since the one-sided limits are not equal, the two-sided limit does not exist. **e) $\lim_{x \to -\infty} f(x)$** - For large negative $x$, $f(x)$ is constant at 4. - So, $\lim_{x \to -\infty} f(x) = 4$. **f) $\lim_{x \to \infty} f(x)$** - For large positive $x$, $f(x)$ approaches the horizontal asymptote $y = -2$ from above. - So, $\lim_{x \to \infty} f(x) = -2$. **Final answers:** $$\lim_{x \to -4^-} f(x) = 4, \quad \lim_{x \to -4^+} f(x) \neq 4, \quad \lim_{x \to -4} f(x) \text{ does not exist}$$ $$\lim_{x \to -2} f(x) = -8$$ $$\lim_{x \to 0} f(x) = 0$$ $$\lim_{x \to 2^-} f(x) = +\infty, \quad \lim_{x \to 2^+} f(x) = -2, \quad \lim_{x \to 2} f(x) \text{ does not exist}$$ $$\lim_{x \to -\infty} f(x) = 4$$ $$\lim_{x \to \infty} f(x) = -2$$