1. **Problem Statement:** Estimate the following limits using the graph of $y=f(x)$: a. $\lim_{x \to 0^-} f(x)$ b. $\lim_{x \to 0^+} f(x)$ c. $\lim_{x \to 0} f(x)$ d. $\lim_{x \to 2^-} f(x)$ e. $\lim_{x \to 2^+} f(x)$ f. $\lim_{x \to 2} f(x)$ g. $\lim_{x \to 4^-} f(x)$ h. $\lim_{x \to 4^+} f(x)$ i. $\lim_{x \to 4} f(x)$ 2. **Analyzing each limit using the graph:** a. $\lim_{x \to 0^-} f(x)$: As $x$ approaches 0 from the left, the function approaches $0$ (the curve reaches $y=0$ at $x=0$ but the filled dot is at $y=1$). b. $\lim_{x \to 0^+} f(x)$: As $x$ approaches 0 from the right, the function value starts at $1$ (an open circle) but the limit concerns the values approaching from the right side. The graph shows a point at $y=1$ (open circle) so the limit from right is also close to $1$. c. $\lim_{x \to 0} f(x)$: The left and right limits are $0$ and $1$ respectively, which are not equal, so the limit at $x=0$ does not exist. d. $\lim_{x \to 2^-} f(x)$: As $x$ approaches 2 from the left, the function value is approaching $-2$ (filled dot at $y=-2$ at $x=2$). e. $\lim_{x \to 2^+} f(x)$: Approaching 2 from the right, the function comes from an open circle at $y=0$ and moves upward, so the limit from the right is $0$. f. $\lim_{x \to 2} f(x)$: Since the left and right limits ($-2$ and $0$) differ, the limit at $x=2$ does not exist. g. $\lim_{x \to 4^-} f(x)$: Approaching 4 from the left, the function is coming up from around $5$ (open circle at $y=5$ at $x=4$). h. $\lim_{x \to 4^+} f(x)$: The function value at $x=4$ is a filled dot at $4$, but the limit from the right is not defined since the graph ends (or the function jumps), so we estimate the right-hand limit as $4$ (value of the filled dot). i. $\lim_{x \to 4} f(x)$: Left and right limits are $5$ and $4$ respectively, not equal, so the limit at $x=4$ does not exist. 3. **Summary:** $$\begin{aligned} &\lim_{x \to 0^-} f(x) = 0 \\ &\lim_{x \to 0^+} f(x) = 1 \\ &\lim_{x \to 0} f(x) \quad \text{does not exist} \\ &\lim_{x \to 2^-} f(x) = -2 \\ &\lim_{x \to 2^+} f(x) = 0 \\ &\lim_{x \to 2} f(x) \quad \text{does not exist} \\ &\lim_{x \to 4^-} f(x) = 5 \\ &\lim_{x \to 4^+} f(x) = 4 \\ &\lim_{x \to 4} f(x) \quad \text{does not exist} \end{aligned}$$ Each limit estimate is based on the values the function approaches from the left and right sides of the point.