Understand the true meaning of limits

[Taturana]

I'm trying to understand the true meaning of limits. But a doubt occurred me...

Limits try to tell us the behavior of a function near a particular input, right? Ok...

So:

\lim_{x \to 0^{+}}{1\over x} = \infty
\lim_{x \to \infty}{1\over x} = 0

These two limits are true, right? Ok...

So, by the first limit we can understand that when x tends to 0, 1/x tends to infinity... x will never be zero and 1/x will never be infinity, but for us to get this result in the limit we SUPPOSE that if x would be 0 when 1/x would be infinity. The same thing for the other limit. Am I right? If yes...

So then a limit is a supposition, limits are not "real" (?) because they're a supposition. So what happens to all the things discovered using limits? Are they suppositions too?

Derivatives are defined as a limit when deltaX goes to 0. How can the derivative represent the REAL slope of the line if limits are not real (are only supposition)?

Maybe I was doing a lot of confusion with these concepts, could someone give me a light please?

Thank you,
Rafael Andreatta

 

[Mark44]

I'm trying to understand the true meaning of limits. But a doubt occurred me...

Limits try to tell us the behavior of a function near a particular input, right? Ok...

So:

\lim_{x \to 0}{1\over x} = \infty
\lim_{x \to \infty}{1\over x} = 0

These two limits are true, right?

No, the first limit doesn't exist in any sense. As x approaches zero from the right, 1/x gets larger without bound. As x approaches 0 from the left, 1/x gets more and more negative without bound.

Ok...

So, by the first limit we can understand that when x tends to 0, 1/x tends to infinity... x will never be zero and 1/x will never be infinity, but for us to get this result in the limit we SUPPOSE that if x would be 0 when 1/x would be infinity. The same thing for the other limit. Am I right? If yes...

Again, no. Let's look at the other limit you wrote. The larger x gets, the closer x gets to 0. True, there is no finite value of x for which 1/x is equal to zero, but the purpose of limits is to observe the behavior of a function at a point where you can't just evaluate the function. You can't just substitute infinity in the expression 1/x, so limits give us a way to see what's happening in some region close to a number (in the case of a limit at a point of discontinuity) or for very large or very negative values of x, as in this case.

With limits, we don't much care about the value of a function or expression at a particular point, but rather, near that same point. For example, we can find that the derivative of f(x) = x^2 at (1, 1) is 2. If you use the limit definition of the limit, the difference quotient gives you (x^2 - 1)/(x - 1). Obviously you can't evaluate this expression at x = 1, but if you evaluate it for numbers close to 1 (either smaller than 1 or larger than 1), the closer your number is to 1, the closer the rational expression is to 2.

So then a limit is a supposition, limits are not "real" (?) because they're a supposition. So what happens to all the things discovered using limits? Are they suppositions too?

Derivatives are defined as a limit when deltaX goes to 0. How can the derivative represent the REAL slope of the line if limits are not real (are only supposition)?

Maybe I was doing a lot of confusion with these concepts, could someone give me a light please?

Thank you,
Rafael Andreatta

 

[Taturana]

Thank you for the reply Mark44.

But you didn't answer my main question that was: if limits are suppositions (so they are not real), how could things derived from limits be real (like derivatives)?

 

[Hurkyl]

No, the first limit doesn't exist in any sense. As x approaches zero from the right, 1/x gets larger without bound. As x approaches 0 from the left, 1/x gets more and more negative without bound.

The first limit he wrote is the limit as x approaches from the right.

 

[Taturana]

The first limit he wrote is the limit as x approaches from the right.

Yeh but I edited the post, in the first post I forgot the + sign above zero...

 

[Hurkyl]

but for us to get this result in the limit we SUPPOSE that if x would be 0 when 1/x would be infinity.

No, limits are about describing the behavior of a function as the argument approaches a number.

The only time this should have any relationship to the value of the function at that number is when the function is continuous at that number.

 

[Taturana]

No, limits are about describing the behavior of a function as the argument approaches a number.

The only time this should have any relationship to the value of the function at that number is when the function is continuous at that number.

Okay, so limits are SUPPOSITIONS, right? Limits tells us what happens to the value of function when argument APPROACHES a number.

How could then limits be used to represent real things (that are not suppositions) like derivatives? (In derivatives we obtain the REAL slope and not a supposition)

 

[Hurkyl]

Okay, so limits are SUPPOSITIONS, right? Limits tells us what happens to the value of function when argument APPROACHES a number.

No, the limit is not a supposition. The limit tells us that the function really does approach a number.

How could then limits be used to represent real things (that are not suppositions) like derivatives? (In derivatives we obtain the REAL slope and not a supposition)

If we are interested in a number, and we know some function approaches that number, then computing the limit tells us the number we are interested in.

 

[Taturana]

No, the limit is not a supposition. The limit tells us that the function really does approach a number.

But not necessarily reaches that number, right? (We cannot plug 0 on 1/x to get infinity or minus infinity)

Do you understand my point?

 

[Hurkyl]

But not necessarily reaches that number, right? (We cannot plug 0 on 1/x to get infinity or minus infinity)

Do you understand my point?

Yes, and it doesn't make sense. (although it is an unfortunately common misunderstanding)

What does "reaching" have to do with anything? (What do you really mean by "reaching" anyways?)

If:

• I want to know A.
• I know f(x) approaches A as x approaches 0
• I have computed that L is the limit of f(x) as x approaches 0

then I conclude

A = L.​

 

[Taturana]

Yes, and it doesn't make sense. (although it is an unfortunately common misunderstanding)

What does "reaching" have to do with anything? (What do you really mean by "reaching" anyways?)

If:

• I want to know A.
• I know f(x) approaches A as x approaches 0
• I have computed that L is the limit of f(x) as x approaches 0

then I conclude

A = L.​

Sorry, but I'll insist on my doubt. And thank you for the replies Hurkyl.

Using your example: what I mean by reaching is pluging 0 on f(x).

I accept that A and L are the same number but if I compute f(0) I won't necessarily get A, do you understand what I mean? That's why I thought about limits as non-real things.

But I think that's one of the things that you need to "accept" and not understand, dunno...

If someone have more explanations on this, please post...

Thank you

 

[Hurkyl]

Can you explain why you think f(0) matters?

 

[Mark44]

Taturana, you should drop your insistence that limits are suppositions to conclude that limits aren't real, and that therefore, applications of limits (such as the derivative) aren't real. Dividing concepts up into "real things" and "suppositions" is not very useful in this context, since it would effectively wipe out most of calculus.

The most useful aspect of limits is to provide a mechanism whereby we can get a sense of the behavior of a function at a point where the function is undefined. Consider for example the function f(x) = (x² - 1)/(x - 1). This function is defined everywhere except at x = 1. An obvious question would be, since we can't evaluate f(1), can we determine some value that f(x) approaches when x is near 1?

The way we do that is with limits, which we can use to show that
\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2

Using the epsilon-delta definition of the limit, it is possible for one person to specify how close to 2 f(x) should be, and the other person to specify how close to 1 x should be so that f evaluated at any value in that interval around 1 (excluding 1 itself) produces a value that is within the desired closeness to 2.

If you want f(x) to be within .01 of 2, I can say how close x needs to be to 1 so that |f(x) - 2| < .01. If that's still not good enough for you, and you want f(x) to be within .0001 of 2, I can find an interval around 1 so that all of those x-values (again, excluding 1 itself) satisfy |f(x) - 2| < .0001. I can continue this process until you get tired of it or are convinced that the limit is actually 2, whichever comes first.