2.2.1To exist or not to exist?

Introduction to Limits as Predictions

2-42.

FOR SALE, Part One

Jacinda has a 1988 Rustang that she wants to sell. Travis is interested in buying her car, but they have not decided on a price. Travis offers $$1000$ for the car stating that this is what the car is worth according to its Blue Book value. Jacinda states, “My car is worth more than $\it{$1000}$ ! If I wanted the Blue Book value, I would have traded it in when I bought my new car. If you look at the used cars advertised in the classified ads, you will see it is worth a lot more than $\it{$1000}$ .”

Taking on the challenge, Travis agrees to look at similar Rustangs in the classified section of the newspaper. Below are all the Rustangs that Travis finds advertised.

Year	1978	1980	1981	1983	1984	1986
Asking Price	$$900$	$$1220$	$$1380$	$$1700$	$$1860$	$$2180$

From the data, can you make a prediction about the asking price for a 1988 Rustang? How reliable is this prediction?
Jacinda decides to do her own investigation using a local paper. Below is her data. According to her research, what price do you predict for a 1988 Rustang?
Year
1990
1991
1993
1994
1996
Asking Price
$$4450$
$$5125$
$$6475$
$$7150$
$$8500$
Based on this information, will Travis and Jacinda agree on the price?
Jacinda and Travis decide that additional research is necessary. They grab another paper and find a 1987 Rustang for sale for $$2340$ and a 1989 Rustang for sale for $$3775$ . Will this new information help them to make a decision about the fair price of the car?

2-43.

FOR SALE, Part Two

By trying to predict the price for the 1988 Rustang, we are seeking a “limit,” or a final prediction of the price as the year approaches 1988. This can be written:

$\lim\limits_ { t \rightarrow 1988 ^ { - } } ( \text { asking price } ) = \$ 2500$ and $\lim\limits_ { t \rightarrow 1988 ^ { + } } ( \text { asking price } ) = \$ 3100$

The left-hand limit is read “As the year approaches 1988 from the left, the asking price approaches $\it$2500$ .”

Translate the right-hand limit into a sentence.
$\lim\limits_ { t \rightarrow 1988 } ( \text { asking price } )$ uses both sides of 1988 to estimate a value.
Since $\lim\limits_ {t \rightarrow 1988 ^ { - } } ( \text { asking price } ) \neq \lim\limits_ { t \rightarrow 1988 ^ { + } } ( \text { asking price } )$ we state that the $\lim\limits_ { t \rightarrow 1988 } ( \text { asking price } )$ does not exist because the two sides do not agree.

What must be true about the left- and right-hand limits for the $\lim\limits_ { t \rightarrow 1988 } ( \text { asking price } )$ to exist?

2-44.

STICKY LIMITS

Holly is trying to predict the value of $y$ when $x = 2$ . Unfortunately, her brother Max stuck gum on the area she was trying to look at! Can she still make a good prediction? Estimate $\lim\limits _ { x \rightarrow 2 } f ( x )$ .

Continuous piecewise, left curve, coming from upper left, decreasing to the point (1, comma 1), opening down, right curve, increasing from the point (1, comma 1), opening down, continuing up & right, with splotch on the curve, making the curve not visible between the approximate points, (1.5, comma 2.5), & (2.25, comma 3.25).

2-45.

Since a limit is a prediction based on a pattern of $y$ -values on a continuous graph, will the limit from problem 2-44 change if you found out that $f (2) = 8$ ? Why or why not?

Math Notes

An Intuitive Definition of Limit

When you graph a function $y=f(x)$ , most of the time you can guess what the value of, say, $f(3)$ is by knowing the values of $f(x)$ when $x$ is very close to $3$ . One way to think about this is to assume you have the graph $y=f(x)$ for $2<x<4$ , except at $x=3$ . Can you make a reasonable accurate guess as to the value of $f(3)$ ? If so, and this value is $L$ , we say that the limit of $f(x)$ exists at $x=3$ and use the notation $\lim\limits _ { x \rightarrow 3 } f ( x ) = L$ .

For example, if $g(3.01) = 4.02, g(3.001) = 4.005$ , and $g(2.999) = 3.997$ , it is reasonable to guess that $g(3) = 4$ and therefore $\lim\limits_ { x \rightarrow 3 } g ( x ) = 4$ .

You can also take one-sided limits using numbers less than $a$ (the notation is $\lim\limits_ { x \rightarrow a ^ { - } } f ( x )$ ) or greater than $a$ (the notation is $\lim\limits_{ x \rightarrow a ^ { + } } f ( x )$ ).

An important point is that $\lim\limits_ { x \rightarrow a } f ( x )$ does not need to equal $f (a)$ .

2-46.

Express the following limit statements as approach statements using complete sentences. Then draw graphs that can represent each limit.

$\lim\limits _ { x \rightarrow 5 } f ( x ) = 6$

$\lim\limits _ { h \rightarrow 3 ^ { + } } g ( h ) = - \infty$

2-47.

Without a calculator, sketch $y = \sqrt{x-1} - 2$ .

Write a complete set of approach statements for this function. Include $x \rightarrow 1^+$ .
Approach statements describe what $y$ is approaching as $x$ approaches some value. This is the same as a limit. For example, one approach statement for $y = \sqrt{x-1} - 2$ can be rewritten using limits as:
$\lim\limits_ { x \rightarrow 1 ^ { + } } ( \sqrt { x - 1 } - 2 ) = - 2$
Use your approach statement from part (a) to rewrite $x \rightarrow 1^−$ and $x \rightarrow \infty$ as limit statements.

Compute without a calculator

2-48.

Determine if each of the following conjectures is always true, sometimes true, or never true. Then provide examples and/or counterexamples to support your claim.

Conjecture 1: When a limit exists at a certain $x$ -value, the function is defined for that $x$ -value.

Conjecture 2: If a function is defined at a certain $x$ -value, then the limit exists at that $x$ -value.

2-49.

Translate the following limit equations using a complete sentence. Then draw a graph to represent each situation. Homework Help ✎

$\lim\limits _ { x \rightarrow - 1 ^ { + } } ( \sqrt { x + 1 } + 3 ) = 3$
$\lim\limits_{\text{time}\rightarrow\infty}$ (a soda’s temperature) $=$ room temperature

2-50.

Consider the functions $f(x)=\log(3-x)$ and $g(x) = \sqrt {x-3} -2$ . 2-50 HW eTool (Desmos). Homework Help ✎

What is the domain of each function?
Graph $h ( x ) = \left\{ \begin{array} { l l } { \operatorname { log } ( 3 - x ) } & { \text { for } x < 3 } \\ { \sqrt { x - 3 } - 2 } & { \text { for } x \geq 3 } \end{array} \right.$ on your graphing calculator.
Explain why $h$ is not continuous at $x = 3$ .
What is the range of each function?

2-51.

Sketch a graph of each of the functions below. Compare the equations and their graphs. Then write a complete set of approach statements for each. Homework Help ✎

$y = \frac { ( x + 6 ) ( x - 1 ) } { x - 1 }$

$y = \frac { ( x + 6 ) ( x - 1 ) } { x - 2 }$

Explain why one graph has a hole while the other has a vertical asymptote.
What is the end behavior of each function?

2-52.

Write as many limit statements as you can about the function graphed at right as $x \rightarrow −1$ and $x \rightarrow ±\infty$ . Homework Help ✎

Piecewise, left curve, coming from left below x axis, opening down, ending at open point (negative 1, comma negative 1 half), right curve, starting at closed point (negative 1, comma 1 half, turning at the point (0, comma 1), changing from opening down to opening up at the point (1, comma 1 half, continuing to the right above the x axis

2-53.

Write the equation of the line that passes through the vertex of $y = 2x^2 + 6x − 20$ with a slope of $-\frac { 7 } { 3 }$ . Write your answer in point-slope form. Homework Help ✎

2-54.

If $f(x) = \sqrt [ 3 ] { x }$ , write an expression for each of the following function operations. Homework Help ✎

$f^{−1}(x)$
$f( f^{−1}(x))$
$f^{−1}( f(x))$

2-55.

For $f (x) = \sin(x)$ , an estimation of the area under the curve for $0\le x\le\pi$ is shown at right using six midpoint rectangles of equal width. Homework Help ✎

Estimate the area using these rectangles.
If the shaded region is rotated about the x-axis, then each of these rectangles becomes what shape? Sketch a picture representing this situation.
Estimate the volume of this rotated region by calculating the volume of each of the rotated rectangles.

Periodic curve, x axis scaled from 0 to pi, with visible turning point at (1 half pi, comma 1), & 6 equal width shaded vertical bars, bottom edges on x axis, left bottom vertex of first bar, on the origin, right bottom vertex of sixth bar on the point, (pi, comma 0), with the top edge midpoint of each bar, on the curve.

2-56.

Zuhaib is anxiously waiting for the results of his calculus test and is pacing back and forth as shown in the graph below. Homework Help ✎

First quadrant curve, x axis labeled time, seconds, y axis labeled distance, feet, with approximate points as follows: starting at, (0, comma 1.8), rising & turning right at (4, comma 6), turning down at (7, comma 6), falling & turning at (9, comma 1), ending at (12, comma 3), with highlighted & labeled points: a, (2, comma 4), b, (5, comma 6), c, (8, comma 4), d, (11, comma 2).

At which point $(a, b, c, \text{ or } d)$ is Zuhaib’s speed the greatest? Approximate the rate.
At which point is Zuhaib’s velocity the greatest? Approximate the rate.
What is the difference between speed and velocity?

Year	1990	1991	1993	1994	1996
Asking Price	$$4450$	$$5125$	$$6475$	$$7150$	$$8500$

Calculus Third Edition

An Intuitive Definition of Limit