1. The problem asks to find the left-hand limit, right-hand limit, two-sided limit, and function value at $x = -2$ for the function $f(x)$ based on the graph. 2. Recall the definitions: - The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left. - The right-hand 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 the left-hand and right-hand limits are equal. - The function value $f(a)$ is the value of the function at $x=a$. 3. From the graph description: - As $x$ approaches $-2$ from the left, the curve goes down sharply towards negative infinity, so $$\lim_{x \to -2^-} f(x) = -\infty.$$ - As $x$ approaches $-2$ from the right, the graph has a vertical asymptote extending downwards to negative infinity, so $$\lim_{x \to -2^+} f(x) = -\infty.$$ - Since both one-sided limits are equal, $$\lim_{x \to -2} f(x) = -\infty.$$ - The function value $f(-2)$ is not explicitly given, but since there is a vertical asymptote at $x=-2$, the function is not defined there, so $$f(-2) \text{ is undefined.}$$ 4. Summary: $$\lim_{x \to -2^-} f(x) = -\infty,$$ $$\lim_{x \to -2^+} f(x) = -\infty,$$ $$\lim_{x \to -2} f(x) = -\infty,$$ $$f(-2) \text{ is undefined.}$$