Graphical Idea of a Limit

YOU WILL LEARN ABOUT:

How to find the limit of a function using its graph.

Finding a limit graphically is used almost every day, especially whenever trends and patterns are analyzed to find an expected value.

What is the expected population a few years from now? Limits on a graph of a certain population give insight into population trends over time, something of interest to governments at all levels.

What is the anticipated effect of deep space travel on the human body? Finding limits on a graph relating the force of gravity to blood pressure helps scientists know what effect being in space will have on the human heart.

What value of an investment is expected at some future date? Interpreting a graph using limits allows financial analysts to predict future values for, say, the purpose of saving up for a purchase or retirement.

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GRAPHICAL IDEA OF A LIMIT

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f (x) = 1.0000

x = -6.0000

$f (x) = \{\begin{cases} - 0.5 x - 2 & x \leq - 1 \\ 2 - x^{2} & - 1 < x < 1 \\ 2 & x = 1 \\ 2 - x^{2} & 1 < x \leq 2 \\ 0.1 {(x - 2)}^{3} - 2 & x \geq 2 \end{cases}$

$\begin{matrix} f (x) = \frac{x^{2}}{x^{2} - 1} \end{matrix}$

$\begin{array}{c} f (x) = \sin (\frac{1}{x}) \end{array}$

〈 0.706,0.708 〉

〈 , 〉

Function 1

Function 2

Function 3

1 2 3

1. The limit of f (x) as

may be different than f (c). For what values of c does the limit of f (x) as

exist but not equal f (c)? Check all that apply.

c = -1

c = 0

c = 1

c = 2

2. When the limit of f (x) as

does equal f (c) , the function is said to be continuous at c. At which of the following values of c is f (x) continuous? Check all that apply.

c = -1

c = 0

c = 1

c = 2

3. The limit of f (x) as

does not exist for which of the following values of c? Check all that apply.

c = -1

c = 0

c = 1

c = 2

1 2 3

1. Slowly move x toward 1 from the left. As

, we say that there is no expected value of f (x), so the limit of f (x) as

does not exist. Why?

f (x) rises higher and higher (that is, higher up the y-axis) as .

f (x) falls farther and farther (that is, lower down the y-axis) as .

f (x) approaches 1 from the left but not from the right.

f (x) approaches 1 from the right but not from the left.

2. Slowly move x toward 1 from the right. As

, we say that there is no expected value of f (x), so the limit of f (x) as

does not exist. Why?

f (x) rises higher and higher (that is, higher up the y-axis) as .

f (x) falls farther and farther (that is, lower down the y-axis) as .

f (x) approaches 1 from the left but not from the right.

f (x) approaches 1 from the right but not from the left.

3. Move x toward the far right. What happens to the value of f (x) as x gets bigger and bigger? That is, does the limit of f (x) as

exist, and if so, what is it?

The limit exists because f (x) approaches 0 as .

The limit exists because f (x) approaches 1 as .

The limit does not exist because f (x) keeps continues to rise as .

The limit does not exist because f (x) continues to fall as .

1 2 3

1. Move x toward 0 slowly from the left. What is the behavior of f (x)?

f (x) rises higher and higher up.

f (x) falls farther and farther down.

f (x) repeatedly rises and falls.

None of the above.

2. Move x toward 0 slowly from the right. What is the behavior of f (x)?

f (x) rises higher and higher up.

f (x) falls farther and farther down.

f (x) repeatedly rises and falls.

None of the above.

3. Which of the following statements is true?

None of the above