Negation of Limit: Am I Right? What Am I Doing Wrong?

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In summary, the correct negation of the definition of a limit is "for some epsilon, there is no delta such that for all x, if 0<|x-a|<delta, then |f(x)-L|<epsilon." This means that there exists at least one value of epsilon for which there is no corresponding delta that satisfies the definition. This is different from continuity, where the function must also be defined at the limit point. The function 1/(x+1) is continuous everywhere on its domain, but if f(-1) is not defined, it does not affect the continuity of the function. The negation of a finite limit is "there exists an epsilon for which there is no delta such that for
  • #1
soulflyfgm
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so is that a good negation of the defenition of the limit?

A function f with domain D doesn't not have limit L at a point c in D iff
not for every number E > 0 there is a corresponding number G >0 such if |F(x) - L| <E then is not the case 0< |x-a|<G
am i right? wat am i doing wrong?
thx so much. I made a new post that way ppl won't get confuse with the other post.
thank u so much
 
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  • #2
I am guessing that the statement you want to negate is something like "for each e there is a d such that X(d) implies Y(e)." So the negation should be "for some e there is not any d such that X(d) implies Y(e)."
 
  • #3
It's easy if you use quantifiers:
Definition of [itex]\lim_{x\to a}f(x)=L[/itex]:

[tex]\forall \epsilon>0 \exists \delta>0 : |x-a|<\delta \Rightarrow |f(x)-L|<\epsilon[/tex]

To negate this, simply use the rules:
[tex]\neg (\forall x:P) \iff \exists x: \neg P[/tex]
[tex]\neg (\exists x:P) \iff \forall x : \neg P[/tex]
 
  • #4
is this right?

is this the right negation of the statement above?

[tex]\exists\epsilon>0 \forall \delta>0 : |x-a|<\delta \wedge\|f(x)-L|\geq\epsilon[/tex]

i am also using this fact
~(P=>Q) = P^~Q

how can i illustrate this negation? would it be a function that is not continuous at a point such as f(x) = 1/(x+1)?
thank you very much for al ur help!
 
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  • #5
I always seem to forget that f(a) isn't important in the definition.
The function doesn't even have to be defined at the limit point. The correct definition is:

[tex]\forall \epsilon>0 \exists \delta>0 : 0<|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon[/tex]

So change [itex]|x-a|<\varepsilon[/itex] to [itex]0<|x-a|<\varepsilon[/itex], then it's correct.

This is different from continuity! A function is continuous at a if [itex]\lim \limits_{x\to a}f(x)=f(a)[/itex], which says 3 things:
1. The limit exists
2. f(a) is defined
3. The 2 are equal.

More precisely, a function is continuous at x=a if
[tex]\forall \epsilon>0 \exists \delta>0 : |x-a|<\delta \Rightarrow |f(x)-f(a)|<\epsilon[/tex]

The function 1/(x+1) is perfectly continuous everwhere on its domain. The point f(-1) is not defined so it's no problem. If you define f(-1)=0, then it's not continuous anymore.
 
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  • #6
Is this the right negation of a finite limit?

[tex]
\neg(\lim_{x\to a}f(x)=L) \iff \exists \epsilon>0 \forall \delta>0\exists x: 0<|x-a|<\delta \Rightarrow |f(x)-L|\geq\epsilon \vee \neg\exists f(x) [/tex]

Thanks.
 

1. What does "negation of limit" mean?

The negation of limit refers to the concept in mathematics where the limit of a function does not exist. This means that the function does not approach a specific value as the independent variable approaches a certain value.

2. How do I know if I am right in negating a limit?

To determine if you are right in negating a limit, you can use the definition of a limit and check if the function does not approach a specific value as the independent variable approaches a certain value. You can also graph the function to visually see if the limit does not exist.

3. Can I negate any limit?

Yes, you can negate any limit as long as the function does not approach a specific value as the independent variable approaches a certain value. However, some limits may be more difficult to negate than others and may require more advanced mathematical techniques.

4. What are some common mistakes in negating a limit?

One common mistake in negating a limit is forgetting to check if the function is approaching a specific value as the independent variable approaches a certain value. Another mistake is incorrectly using algebraic manipulations to negate the limit, as this may change the behavior of the function.

5. How can I improve my skills in negating limits?

To improve your skills in negating limits, it is important to have a strong understanding of the definition of a limit and the properties of limits. Practice solving different types of limit problems and seek help from a math tutor or teacher if needed. It may also be helpful to review theorems and techniques specific to negating limits.

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