Explain what a limit means

[mileena]

Homework Statement

Explain in your own words what:

lim f(x) = 5
x→2

means.

Homework Equations

None

The Attempt at a Solution

I wrote:

"As x approaches 2, the limit of the function f(x) is 5 (in other words, f(x) approaches 5). The value of the function that is limited can be different than the limit of the function itself."

But I do not know if there can be a function value but with an undefined limit. I also want to say above that a function value may or may not be the same as the limit, and that either value may be undefined. But I don't know for sure if that is true.

For example, what is:

lim f(x) if f(x) = x (in other words, if y = x)
x→1

Above we have a straight line graph with a slope of 1, and a y-intercept of 0.

But is the limit really 1? Even though the graph goes right through f(x) = 1?

 

[Simon Bridge]

You started out just writing in words what the notation says ... which is not what they want you to do.
You need to explain what "the limit of the function of x is five as x approaches 2" actually means.

You are correct, the limit does not mean that the function takes the value of 5 when x=2. It may do, but it doesn't have to. I think you need to do some more reading.

http://www.mathsisfun.com/calculus/limits-formal.html
... the first example is like what you were talking about.

 

[mileena]

Thank you so much for replying!

You are right about that first example. I learned that the limit can be undefined at a particular x value, but the function can have a value at that same x-value. And vice-versa: so the function can be undefined at a particular x-value, but there still can be a limit for that same x. Or both the limit and function can be undefined for a particular ^x, or both can be the same value!

This is what I revised my answer to (plus there was another short question at the end):

"In your own words:

As x approaches 2, the limit of the function f(x) is 5 (in other words, f(x) approaches 5, and may or may not equal 5). The value of the function with a limit can be different than the limit of the function itself.

Is it possible for the above equation to be true and yet f(2) = 3? Explain.

Yes, since the value of the function does not have to equal the value of the limit, although it can. It depends on the coordinate values on the graph of the function. The limit value and the function value are two separate things."

 

[Simon Bridge]

Better - try to explain without using the word "limit".
Did you read the link in post #2?

 

[vela]

"In your own words:

As x approaches 2, the limit of the function f(x) is 5 (in other words, f(x) approaches 5, and may or may not equal 5).

As Simon noted, the part in red is simply reading what the notation $\lim_{x \to 2} f(x)=5$ says. You should just get rid of it.

What exactly does it mean to say that "f(x) approaches 5"? Try to be more specific and precise.

The value of the function with a limit can be different than the limit of the function itself.

What is the phrase function with a limit supposed to mean? I take it what you're saying is that f(2) doesn't necessarily have to be equal to 5, which is what you also said above, right?

Is it possible for the above equation to be true and yet f(2) = 3? Explain.

Yes, since the value of the function does not have to equal the value of the limit, although it can. It depends on the coordinate values on the graph of the function. The limit value and the function value are two separate things."

What do you mean by the sentence in red? Why are f(2) and the limit separate things? What's the fundamental reason they can be different?

 

[CompuChip]

Just for your own understanding, you may also want to look at the similarities and differences between for example

$f(x) = x + 1$ (as $x \to 0$)

$g(x) = \begin{cases} x + 1 & \text{ if } x \neq 0 \\ -1 & \text{ if } x = 0 \end{cases}$ (as $x \to 0$)

$h(x) = 1/x$ (as $x \to 0$)

 

[mileena]

Ok, thanks you everyone for your help!

I took your recommendations, and got rid of the word "limit" in my definition of limit! That makes a lot of sense. :tongue:

I am not going to go too much further as the homework is due in a few hours, and this is supposed to be an easy question. I just took things too far in my (attempted!) quest for perfectionism (although I have so much to learn about calculus that I am just grateful I have an answer here).

This is what I wrote:

In your own words:

As x approaches 2, the value that the function gets close to (i.e., what is known as the limit of the function f(x)) is 5. In other words, as x approaches 2, f(x) approaches 5. Furthermore, the function itself, f(x), may or may not equal 5. The value of the function can be different than the limit of the function itself. f(x) also may or may not continue past 5, with the graph being either continuous or discontinuous at this point.

Is it possible for the above equation to be true and yet f(2) = 3? Explain.

Yes, since the value of the function does not have to equal the value of the limit, although it can. In this case, the value of the function at 2 equals 3, whereas the limit of the function as x approaches 2 is 5. Whether or not the value of the function at x matches the value of the limit of that function as that x is approached depends on the values of the function. For example, if the graph of the function is discontinuous at the limit itself, and the function has a value at some other point, then the function limit will not equal the function value. The limit value and the function value are two separate things.

 

[verty]

Ok, thanks you everyone for your help!

I took your recommendations, and got rid of the word "limit" in my definition of limit! That makes a lot of sense. :tongue:

I am not going to go too much further as the homework is due in a few hours, and this is supposed to be an easy question. I just took things too far in my (attempted!) quest for perfectionism (although I have so much to learn about calculus that I am just grateful I have an answer here).

This is what I wrote:

In your own words:

As x approaches 2, the value that the function gets close to (i.e., what is known as the limit of the function f(x)) is 5. In other words, as x approaches 2, f(x) approaches 5. Furthermore, the function itself, f(x), may or may not equal 5. The value of the function can be different than the limit of the function itself. f(x) also may or may not continue past 5, with the graph being either continuous or discontinuous at this point.

Is it possible for the above equation to be true and yet f(2) = 3? Explain.

Yes, since the value of the function does not have to equal the value of the limit, although it can. In this case, the value of the function at 2 equals 3, whereas the limit of the function as x approaches 2 is 5. Whether or not the value of the function at x matches the value of the limit of that function as that x is approached depends on the values of the function. For example, if the graph of the function is discontinuous at the limit itself, and the function has a value at some other point, then the function limit will not equal the function value. The limit value and the function value are two separate things.

This seems fine to me. The one quibble I have is about the phrase "if the graph of the function is discontinuous at the limit itself". Whether the graph is continuous or not is related to not lifting your pencil. Arguing from geometry is probably not what the teacher wants in this case.