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

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$s(0)=0, s(5)=0$.

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s^{\prime}(3)=-16$$

Find an equation that could be represented by as we discussed earlier.

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s(3)=96$$

For the following exercise,

Find the average rate of change $\frac{f(x+h)-f(x)}{h}$.

$$f(x)=\sqrt{x}$$

Allow the speed of light, $c$, to be equal to $1.0$. If the mass, $m$, is $1$, what occurs to $m$ as $v \rightarrow c$? Using the values listed as we discussed earlier, make a conjecture as to what the mass is as $v$ approaches $1.00$.

$$\begin{array}{|c|c|} \hline \boldsymbol{v} & \boldsymbol{m} \\ \hline 0.5 & 1.15 \\ \hline 0.9 & 2.29 \\ \hline 0.95 & 3.20 \\ \hline 0.99 & 7.09 \\ \hline 0.999 & 22.36 \\ \hline 0.99999 & 223.61 \\ \hline \end{array}$$

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s^{\prime}(2)=16$$

For the following exercise,

Find the average rate of change $\frac{f(x+h)-f(x)}{h}$.

$$f(x)=\frac{1}{x^2}$$

According to the Theory of Relativity, the mass $m$ of a particle depends on its velocity $v$. That is

$$m=\frac{m_o}{\sqrt{1-\frac{v^2}{c^2}}}$$

where $m_o$ is the mass when the particle is at rest and $c$ is the speed of light. Find the limit of the mass, $m$, as $v$, approaches $c^{-}$.

For the following exercise,

Find the average rate of change $\frac{f(x+h)-f(x)}{h}$.

$$f(x)=\frac{1}{x}$$

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s(2)=96$$

For the following exercise,

Explain the notation in words. The volume $f(t)$ of a tank of gasoline, in gallons, $t$ minutes after noon.

$$f(240)=500$$

For the following exercise,

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

Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: $f(x)=\left|\frac{1-x}{x}\right|$ and $g(x)=\left|\frac{1+x}{x}\right|$ as $x$ approaches $0$. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions $f(x)$ and $g(x)$ as $x$ approaches $0$. If the functions have a limit as $x$ approaches $0$, state it. If not, discuss why there is no limit.