How to approach if \lim_{x\to0}\frac{f(x)}{x} = 1 then \lim_{x\to0}f(x) = 0?

Integrate

Integrate

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How to approach if \lim_{x\to0}\frac{f(x)}{x} = 1 then \lim_{x\to0}f(x) = 0?

I really just don't know how to handle If - Then problems.


I always get them wrong.

How should I approach this one?


My thinking is \lim_{x\to0}f(x) can equal anything and still have lim_{x\to0}\frac{f(x)}{x} = 1.
 
Last edited by a moderator:
Integrate said:
I really just don't know how to handle If - Then problems.

I always get them wrong.

How should I approach this one?
Click to expand...
I'm confused. Are you saying you were asked to prove that

if [imath]\lim_{x\to0}\frac{f(x)}{x} = 1[/imath], then [imath]\lim_{x\to0}f(x) = 0[/imath] ?​

Or are you trying to understand what it means? Or to decide whether it is true? Or what? You haven't clearly stated what you are trying to do, or what the context is.

And when you say,
Integrate said:
My thinking is \(\lim_{x\to0}f(x)\) can equal anything and still have \(\lim_{x\to0}\frac{f(x)}{x} = 1\).
Click to expand...
can you give an example in which the former is non-zero but the latter is true?
 
Pardon.


This is a true or false question in the form of an If-Then.

I also have a typo in the original post.

\lim_{x\to0}\frac{f(x)}{x} = 1 then f(x) = 0​

 
What does if \(\displaystyle \lim_{x\to0}\frac{f(x)}{x} = 1\) mean from the formal definition mean?

To do If-Then proofs you need to start with the if part (ie \(\displaystyle \lim_{x\to0}\frac{f(x)}{x} = 1\)) and conclude the then part (\(\displaystyle \lim_{x\to0}f(x) = 0\))
 
Integrate said:
Pardon.


This is a true or false question in the form of an If-Then.

I also have a typo in the original post.

\lim_{x\to0}\frac{f(x)}{x} = 1 then f(x) = 0​

Click to expand...
Did you try using f(x)=x to see what the limit equals?
 
Integrate said:
Pardon.


This is a true or false question in the form of an If-Then.

I also have a typo in the original post.

\lim_{x\to0}\frac{f(x)}{x} = 1 then f(x) = 0​

Click to expand...
That's not a typo; it's a different question, and probably easier. Please try answering both!

You haven't answered my last question: What examples have you tried? That's a very useful way to start this kind of problem. Try some cases where the condition is true, and see if the conclusion is always true. In particular, try to think of an example in which the conclusion will be false.

For example, can you make the limit 1 for a function f(x) that is not identically equal to 0? And can you make the limit 1 for a function f(x) that does not even approach 0?
 
IMG_1309.jpg

How's this?


My "does not work case" is the second one. It makes the limit equal 1 by making x=3 which goes against the original statement.
 
Integrate said:
View attachment 35819

How's this?


My "does not work case" is the second one. It makes the limit equal 1 by making x=3 which goes against the original statement.
Click to expand...
Okay, this is yet a third different question. You started with this:
Integrate said:
if \(\lim_{x\to0}\frac{f(x)}{x} = 1\) then \(\lim_{x\to0}f(x) = 0\)
Click to expand...
There the conclusion was a limit at zero.

Then you tried this:
Integrate said:

[If] \(\lim_{x\to0}\frac{f(x)}{x} = 1\) then \(f(x) = 0\)​

Click to expand...
Here the conclusion was a function equal to 0 everywhere.

Now you say this:

1684367042718.png
Here, it's a function equal to zero at x=0.

But your second example somehow involves x=3, which is irrelevant to any of these three questions! And your last line appears to be taking the implication in the wrong direction.

On the other hand, this example does suggest an interesting angle: Can the condition be true though f doesn't have either a value or a limit at 0?

Until you show us the actual problem, as an image, I'm going to ignore anything more that you say, because this whole thread is so unstable.
 
Integrate said:
How to approach if \lim_{x\to0}\frac{f(x)}{x} = 1 then \lim_{x\to0}f(x) = 0?

I really just don't know how to handle If - Then problems.


I always get them wrong.

How should I approach this one?


My thinking is \lim_{x\to0}f(x) can equal anything and still have lim_{x\to0}\frac{f(x)}{x} = 1.
Click to expand...
Like Dr. Peterson, I find this a frustrating thread. I am going to assume that the problem is

[math]\text {True or false: If} \lim_{x \rightarrow 0} \dfrac{f(x)}{x} = 1, \text { then } f(x) = 0.[/math]
So first what does the question mean?

The given statement is not true if there is ANY example of a function of x for which it is not true. That is, the statement is true only if it is ALWAYS true. A single contrary example is enough to make the statement false even if there are many examples that make it true.

Thus, there are two ways to attack this kind of problem. One is to prove by logic that it always is true; the other is to find an example where it is not true.

Consider [imath]f(x) = x^2 + x.[/imath]

What is [imath]\lim_{x \rightarrow 0} \dfrac{f(x)}{x}.[/imath]

Why?

What is f(0)? What is f(10)?

Is the proposition always true?
 
Suppose f(x) =x
Then f(x)/x = x/x = 1 if x is not 0. So the limit in the problem will equal 1. BUT f(x) is not 0. Done.
 
If the problem says f(x) = 0, it's trivial. If it says f(0) = 0, it's slightly interesting. If it's what the OP said, then it involves significant ideas. That's why I greatly prefer that version, and can't believe either of the others could be correct.

If we had the correct problem, we could be having a useful discussion, like #11 but much deeper.
 
I see everyone is very upset with me.

I apologize for my inability to properly ask a question.

Latex is something I don't really know how to write or read but know that it is the formatting needed for this forum.

Feel free to be annoyed with me because I also am very irritated by my incompetency to even ask a question.

Here is the set of questions I pulled mine from originally.

Screenshot 2023-05-18 133054.png


I believe I was getting (b) and (e) confused and interchanging them in this thread.


Both (b) and (e) I have difficulty understanding how to prove true or false.


Mods, feel free to delete this thread.
 
Thanks for posting the exact problem.
Which part do you need help with? Can we see your work so we can guide you to the solution?
 
Integrate said:
I apologize for my inability to properly ask a question.

Latex is something I don't really know how to write or read but know that it is the formatting needed for this forum.

Feel free to be annoyed with me because I also am very irritated by my incompetency to even ask a question.

Here is the set of questions I pulled mine from originally.

View attachment 35832

I believe I was getting (b) and (e) confused and interchanging them in this thread.

Both (b) and (e) I have difficulty understanding how to prove true or false.
Click to expand...
No, you didn't interchange them; you asked them both, at different times, plus a hybrid that no one would ask.

If you had just given us this image of the actual problem from the start, everything would be fine. THAT'S the way to ask a question. (We would then have asked you to show your thinking on the other parts, so we can see whether your difficulty is specific to (b) and (e), or more general.)

By the way, Latex is not required for this forum! You can type math just fine without it, or use an image as you have finally done. (I'm intentionally not using it here.)

For (b), you need a counterexample where f(0) is not 0, but lim f(x)/x is 1. I've hinted at this: What if f(0) is not defined?

For (e), you can try for a counterexample, and if you don't find one, look at what you have been taught to see if there is a theorem that might imply this. Since we don't know what you know, you'll have to help us out there. Show us what theorems you learned. (Use an image from your textbook, if necessary.)
 
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