§1.2 The notion of limit

Introduction

The Notion of Limit

Instantaneous Velocity

Summary

Suppose that $g$ is the function given by the graph below. Use the graph in Figure 1.2 to answer each of the following questions.

Determine the values $g(-2)$, $g(-1)$, $g(0)$, $g(1)$, and $g(2)$, if defined. If the function value is not defined, explain what feature of the graph tells you this.

For the function $g$ pictured in Figure 1.2 from Preview Activity 1.2, what can we say about the values of

?

For the function $g$ pictured in Preview Activity 1.2, what can we say about the values of

?

For the function $g$ pictured in Preview Activity 1.2, what can we say about the values of

?

For each of the following functions, we'd like to know whether or not the function has a limit at the stated $a$-values. Use both numerical and algebraic approaches to investigate and, if possible, estimate or determine the value of the limit. Compare the results with a careful graph of the function on an interval containing the points of interest.

-$f(x) = \frac{4-x^2}{x+2}$; $a = -1$, $a = -2$-$g(x) = \sin\left(\frac{\pi}{x}\right)$; $a = 3$, $a = 0$

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

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

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

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

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

Determine the instantaneous velocity of the object when $t = 3$. Include units on your answer.

Use the graph to estimate the average velocity of the object on each of the following intervals: $[0.5,1]$, $[1.5,2.5]$, $[0,5]$. Draw each line whose slope represents the average velocity you seek.

How could you use average velocities or slopes of lines to estimate the instantaneous velocity of the object at a fixed time?

Use the graph to estimate the instantaneous velocity of the object when $t = 2$. Should this instantaneous velocity at $t = 2$ be greater or less than the average velocity on $[1.5,2.5]$ that you computed in (a)? Why?

Consider the function whose formula is $f(x) = \frac{16-x^4}{x^2-4}$.

• What is the domain of $f$? - Use a sequence of values of $x$ near $a = 2$ to estimate the value of $\lim_{x \to 2} f(x)$, if you think the limit exists. If you think the limit doesn't exist, explain why. - Use algebra to simplify the expression $\frac{16-x^4}{x^2-4}$ and hence work to evaluate $\lim_{x \to 2} f(x)$ exactly, if it exists, or to explain how your work shows the limit fails to exist. Discuss how your findings compare to your results in (b). - True or false: $f(2) = -8$. Why? - True or false: $\frac{16-x^4}{x^2-4} = -4-x^2$. Why? How is this equality connected to your work above with the function $f$? - Based on all of your work above, construct an accurate, labeled graph of $y = f(x)$ on the interval $[1,3]$, and write a sentence that explains what you now know about $\lim_{x \to 2} \frac{16-x^4}{x^2-4}$.

Let $g(x) = -\frac{|x+3|}{x+3}$.

• What is the domain of $g$? - Use a sequence of values near $a = -3$ to estimate the value of $\lim_{x \to -3} g(x)$, if you think the limit exists. If you think the limit doesn't exist, explain why. - Use algebra to simplify the expression $\frac{|x+3|}{x+3}$ and hence work to evaluate $\lim_{x \to -3} g(x)$ exactly, if it exists, or to explain how your work shows the limit fails to exist. Discuss how your findings compare to your results in (b). (Hint: $|a| = a$ whenever $a \ge 0$, but $|a| = -a$ whenever $a \lt 0$.) - True or false: $g(-3) = -1$. Why? - True or false: $-\frac{|x+3|}{x+3} = -1$. Why? How is this equality connected to your work above with the function $g$? - Based on all of your work above, construct an accurate, labeled graph of $y = g(x)$ on the interval $[-4,-2]$, and write a sentence that explains what you now know about $\lim_{x \to -3} g(x)$.

For each of the following prompts, sketch a graph on the provided axes of a function that has the stated properties.

-$y = f(x)$ such that -$f(-2) = 2$ and $\lim_{x \to -2} f(x) = 1$-$f(-1) = 3$ and $\lim_{x \to -1} f(x) = 3$-$f(1)$ is not defined and $\lim_{x \to 1} f(x) = 0$-$f(2) = 1$ and $\lim_{x \to 2} f(x)$ does not exist. -$y = g(x)$ such that -$g(-2) = 3$, $g(-1) = -1$, $g(1) = -2$, and $g(2) = 3$- At $x = -2, -1, 1$ and $2$, $g$ has a limit, and its limit equals the value of the function at that point. -$g(0)$ is not defined and $\lim_{x \to 0} g(x)$ does not exist.

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

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