For the following exercise,

With the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a$. If the function has a limit as $x$ approaches $a$, state it. If not, discuss why there is no limit.

$$f(x)=\left\{\begin{array}{ll}\frac{1}{x}-3, & \text { if } x \leq 2 \\ x^3+1, & \text { if } x>2\end{array} a=2\right.$$

The amount of money in an account after $t$ years compounded continuously at $4.25 \%$ interest is given by the formula $A=A_0 e^{0.0425 t}$, where $A_0$ is the initial amount invested. Find the average rate of change of the balance of the account from $t=1$ year to $t=2$ years if the initial amount invested is $\$ 1,000.00$.

For the following exercise, evaluate the limits algebraically.

$$\lim _{x \rightarrow 2}\left(\frac{x^2-5 x+6}{x+2}\right)$$

For the following exercise,

Estimate the functional values and the limits from the graph of the function $f$ provided as we discussed earlier.

$$f(-2)$$

Using, estimate $\lim _{x \rightarrow 0} f(x)$.

$$\begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline-0.1 & 2.875 \\ \hline-0.01 & 2.92 \\ \hline-0.001 & 2.998 \\ \hline 0 & \text { Undefined } \\ \hline 0.001 & 2.9987 \\ \hline 0.01 & 2.865 \\ \hline 0.1 & 2.78145 \\ \hline 0.15 & 2.678 \\ \hline \end{array}$$

For the following exercise,

Use the definition of derivative $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

$$f(x)=3 x-4$$

For the following exercise,

Determine why the function $f$ is discontinuous at a given point on the graph. State which condition fails.

$$f(x)=\frac{x^2-16 x}{x}, a=0$$

For the following exercise, use the graph of $f$ in as we discussed earlier.

At what values of $x$ is $f$ discontinuous? What property of continuity is violated?

The height of a projectile is given by $s(t)=-64 t^2+192 t$ Find the average rate of change of the height from $t=1$ second to $t=1.5$ seconds.

The position functions $s(t)=-16 t^2+144 t$ give the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval $[1,2]$.

What is the left-hand limit of the function as $x$ approaches $0$ as we discussed earlier?

For the following exercise,

The volume $V$ of a sphere with respect to its radius $r$ is given by $V=\frac{4}{3} \pi r^3$.

Find the instantaneous rate of change of V when r = 3 cm.

For the following exercise,

The volume $V$ of a sphere with respect to its radius $r$ is given by $V=\frac{4}{3} \pi r^3$.

Find the average rate of change of V as r changes from $1$ cm to $2$ cm.

What is the right-hand limit of the function as $x$ approaches $0$ as we discussed earlier?