1. **Problem Statement:** We are given a function $G(x)$ graphed as a sinusoidal wave oscillating between $-2$ and $2$ on the interval approximately $[0,5]$. We need to find: (a) $\lim_{x \to 0^-} G(x)$ (b) $\lim_{x \to 0^+} G(x)$ (c) $\lim_{x \to 0} G(x)$ (d) $G(0)$ 2. **Understanding Limits:** - The left-hand limit $\lim_{x \to 0^-} G(x)$ is the value $G(x)$ approaches as $x$ approaches $0$ from values less than $0$. - The right-hand limit $\lim_{x \to 0^+} G(x)$ is the value $G(x)$ approaches as $x$ approaches $0$ from values greater than $0$. - The two-sided limit $\lim_{x \to 0} G(x)$ exists only if both one-sided limits exist and are equal. - The value $G(0)$ is the actual function value at $x=0$. 3. **Analyzing the Graph:** - The graph is defined and oscillates starting at $x=0$ and onwards; it is not defined for $x<0$ (since the domain starts at 0). - Therefore, the left-hand limit $\lim_{x \to 0^-} G(x)$ does not exist because the function is not defined for $x<0$. - The right-hand limit $\lim_{x \to 0^+} G(x)$ can be observed from the graph as $x$ approaches $0$ from the right. - Since the graph is sinusoidal and continuous starting at $x=0$, the right-hand limit equals the function value at $0$. 4. **Reading Values from the Graph:** - At $x=0$, the graph passes through $y=0$ (the origin). - The right-hand limit $\lim_{x \to 0^+} G(x) = 0$. - The function value $G(0) = 0$. 5. **Conclusions:** - (a) $\lim_{x \to 0^-} G(x)$ does not exist (function undefined for $x<0$). - (b) $\lim_{x \to 0^+} G(x) = 0$. - (c) $\lim_{x \to 0} G(x)$ does not exist because the left-hand limit does not exist. - (d) $G(0) = 0$. **Final answers:** $$\lim_{x \to 0^-} G(x) \text{ does not exist}$$ $$\lim_{x \to 0^+} G(x) = 0$$ $$\lim_{x \to 0} G(x) \text{ does not exist}$$ $$G(0) = 0$$