Limit Product 3D893D

1. **State the problem:** We need to find the limit $$\lim_{x \to -3} (g(x) f(x))$$ where $$g$$ and $$f$$ are given piecewise functions graphed around $$x = -3$$. 2. **Understand the limit:** The limit of a product is the product of the limits if both limits exist: $$\lim_{x \to a} (g(x) f(x)) = \left(\lim_{x \to a} g(x)\right) \cdot \left(\lim_{x \to a} f(x)\right)$$ provided both limits exist. 3. **Find $$\lim_{x \to -3} g(x)$$:** - From the graph description, $$g(x)$$ has an open circle at $$(-3, 1)$$ and a closed circle at $$(-3, -1)$$. - The limit depends on the values approaching $$-3$$ from left and right. - From the left, $$g(x)$$ approaches the open circle at $$1$$. - From the right, $$g(x)$$ starts at the closed circle at $$-1$$. - Since left and right limits differ, $$\lim_{x \to -3} g(x)$$ does not exist. 4. **Find $$\lim_{x \to -3} f(x)$$:** - From the graph description, $$f(x)$$ moves from $$(-4, -1)$$ to $$(-3, 0)$$. - So $$\lim_{x \to -3} f(x) = 0$$. 5. **Evaluate the product limit:** - Since $$\lim_{x \to -3} g(x)$$ does not exist, the product limit $$\lim_{x \to -3} (g(x) f(x))$$ does not exist. **Final answer:** The limit does not exist. **Answer choice:** E