Question

officials in state a tend to raise the sales tax in years in which the state faces a budget deficit and then cut the tax when the state has a surplus. the graph to the right shows state a’s state sales tax since it was first established in 1933. let t(x) represent the sales tax per dollar spent in year x. find the following. note that 01 represents 2001.
a. lim t(x)
b. lim t(x)
x→56 x→35⁻
c. lim t(x)
d. lim t(x)
x→35⁺ x→35
e. t(35)
a. select the correct choice below and, if necessary, fill in the answer box within your choice.
○a. lim t(x) = cents
x→56
(round to one decimal place as needed.)
○b. the limit does not exist.
b. select the correct choice below and, if necessary, fill in the answer box within your choice.
○a. lim t(x) = cents
x→35⁻

Explanation:

Step1: Analyze $\lim_{x\to56}T(x)$

As $x$ approaches 56, from the graph, the function value approaches 5. So $\lim_{x\to56}T(x) = 5.0$ cents.

Step2: Analyze $\lim_{x\to35^{-}}T(x)$

As $x$ approaches 35 from the left - hand side, the function value is 3.0 cents. So $\lim_{x\to35^{-}}T(x)=3.0$ cents.

Step3: Analyze $\lim_{x\to35^{+}}T(x)$

As $x$ approaches 35 from the right - hand side, the function value is 4.0 cents. So $\lim_{x\to35^{+}}T(x)=4.0$ cents.

Step4: Analyze $\lim_{x\to35}T(x)$

Since $\lim_{x\to35^{-}}T(x)
eq\lim_{x\to35^{+}}T(x)$, $\lim_{x\to35}T(x)$ does not exist.

Step5: Analyze $T(35)$

The value of the function at $x = 35$ is the value of the function at the solid - dot. From the graph, $T(35)=4.0$ cents.

Answer:

a. A. $\lim_{x\to56}T(x)=5.0$ cents
b. A. $\lim_{x\to35^{-}}T(x)=3.0$ cents
c. A. $\lim_{x\to35^{+}}T(x)=4.0$ cents
d. B. The limit does not exist.
e. $T(35)=4.0$ cents