The notion of limit

Estimate the value of each of the following limits by constructing appropriate tables of values. Then determine the exact value of the limit by using algebra to simplify the function. Finally, plot each function on an appropriate interval to check your result visually.

1. $\displaystyle \lim_{x \to 1} \frac{x^2 - 1}{x-1}$

2. $\displaystyle \lim_{x \to 0} \frac{(2+x)^3 - 8}{x}$

3. $\displaystyle \lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}$

Consider a moving object whose position function is given by $s(t) = t^2$, where $s$ is measured in meters and $t$ is measured in minutes.

1. Determine the most simplified expression for the average velocity of the object on the interval $[3, 3+h]$, where $h \gt 0$.

2. Determine the average velocity of the object on the interval $[3,3.2]$. Include units on your answer.

3. Determine the instantaneous velocity of the object when $t = 3$. Include units on y…

A bungee jumper dives from a tower at time $t=0$. Her height $s$ in feet at time $t$ in seconds is given by $s(t) = 100\cos(0.75t) \cdot e^{-0.2t}+100$.

1. Write an expression for the average velocity of the bungee jumper on the interval $[1,1+h]$.
2. Use computing technology to estimate the value of the limit as $h \to 0$ of the quantity you found in (a).
3. What is the meaning of the value of the limit in (b)? What are its units?