Question

1. did car (a) break the speed limit? what was its velocity, and at what intervals?
2. which of the other cars broke the speed limit? when did the violations happen? (approximate to the nearest.2 seconds and list as intervals.)
3. somewhere near the 4 mile mark, there is a stop sign. did any cars actually stop for it?
4. evaluate the following limits, if they exist.

a. $f(x)=\begin{cases}x^{2}+1 &\text{if }x < - 1\sqrt{x + 1}&\text{if }xgeq - 1end{cases}$ $lim_{x
ightarrow - 1^{-}}f(x)$ and $lim_{x
ightarrow - 1^{+}}f(x)$
b. $lim_{t
ightarrow a}\frac{sqrt{3t + 1}-sqrt{3a + 1}}{t - a}$
c. $lim_{t
ightarrow2}\frac{3t^{2}-7t + 2}{2 - t}$

Explanation:

Step1: Analyze left - hand limit for part a

For $\lim_{x
ightarrow - 1^{-}}f(x)$, since $x
ightarrow - 1^{-}$ means $x < - 1$, we use $f(x)=x^{2}+1$. Substitute $x=-1$ into $x^{2}+1$:
$(-1)^{2}+1=2$.

Step2: Analyze right - hand limit for part a

For $\lim_{x
ightarrow - 1^{+}}f(x)$, since $x
ightarrow - 1^{+}$ means $x\geq - 1$, we use $f(x)=\sqrt{x + 1}$. Substitute $x=-1$ into $\sqrt{x + 1}$:
$\sqrt{-1 + 1}=0$.

Step3: Solve part b

Rationalize the numerator of $\lim_{t
ightarrow a}\frac{\sqrt{3t + 1}-\sqrt{3a+1}}{t - a}$. Multiply by $\frac{\sqrt{3t + 1}+\sqrt{3a + 1}}{\sqrt{3t + 1}+\sqrt{3a + 1}}$:
\[

$$\begin{align*} &\lim_{t ightarrow a}\frac{(3t + 1)-(3a + 1)}{(t - a)(\sqrt{3t + 1}+\sqrt{3a + 1})}\\ =&\lim_{t ightarrow a}\frac{3(t - a)}{(t - a)(\sqrt{3t + 1}+\sqrt{3a + 1})}\\ =&\lim_{t ightarrow a}\frac{3}{\sqrt{3t + 1}+\sqrt{3a + 1}}=\frac{3}{2\sqrt{3a + 1}} \end{align*}$$

\]

Step4: Solve part c

Factor the numerator of $\lim_{t
ightarrow2}\frac{3t^{2}-7t + 2}{2 - t}$. We have $3t^{2}-7t + 2=(3t - 1)(t - 2)$. So the limit becomes $\lim_{t
ightarrow2}\frac{(3t - 1)(t - 2)}{-(t - 2)}=\lim_{t
ightarrow2}-(3t - 1)=-5$.

For questions 1 - 3, without additional information (such as a graph or data about the cars' motion), we cannot answer them.

Answer:

a. $\lim_{x
ightarrow - 1^{-}}f(x)=2$, $\lim_{x
ightarrow - 1^{+}}f(x)=0$
b. $\frac{3}{2\sqrt{3a + 1}}$
c. $-5$