1. **State the problem:** We analyze the sales tax function $T(x)$ at specific years $x$ to find limits and values based on the graph description. 2. **Recall limit definitions:** - The limit from the left $\lim_{x \to a^-} T(x)$ is the value $T(x)$ approaches as $x$ approaches $a$ from values less than $a$. - The limit from the right $\lim_{x \to a^+} T(x)$ is the value $T(x)$ approaches as $x$ approaches $a$ from values greater than $a$. - The two-sided limit $\lim_{x \to a} T(x)$ exists if and only if the left and right limits are equal. - The function value $T(a)$ may differ from the limit if there is a jump or open circle. 3. **Analyze each question:** **a. $\lim_{x \to 57} T(x)$:** - From the graph, for $x < 57$, $T(x) \approx 3$ cents. - Just after 57, $T(x)$ drops to about 2 cents. - Left limit: $\lim_{x \to 57^-} T(x) = 3$ - Right limit: $\lim_{x \to 57^+} T(x) = 2$ - Since left and right limits differ, the two-sided limit does not exist. **b. $\lim_{x \to 67^-} T(x)$:** - For $x$ approaching 67 from the left, $T(x)$ is about 2 cents (from after 57 to 67). - So, $\lim_{x \to 67^-} T(x) = 2$ **c. $\lim_{x \to 67^+} T(x)$:** - For $x$ just greater than 67, $T(x)$ jumps to about 5 cents. - So, $\lim_{x \to 67^+} T(x) = 5$ **d. $\lim_{x \to 67} T(x)$:** - Left limit is 2, right limit is 5, so the two-sided limit does not exist. **e. $T(67)$:** - The graph shows an open circle at 3 cents and filled circles at 5 cents at $x=67$. - The function value corresponds to the filled circle, so $T(67) = 5$ **f. Was 1973 a year of deficit, surplus, or neither?** - From 1967 to 1974, tax is about 5 cents, which is higher than before. - Since taxes are raised when deficit and cut when surplus, and tax is high in 1973, it indicates a deficit year. **Final answers:** $$\lim_{x \to 57} T(x) \text{ does not exist}$$ $$\lim_{x \to 67^-} T(x) = 2$$ $$\lim_{x \to 67^+} T(x) = 5$$ $$\lim_{x \to 67} T(x) \text{ does not exist}$$ $$T(67) = 5$$ 1973 was a year of deficit.