Prove a limit exists using formal definition

This way, it doesn't matter what min{1, ε/3} is. It's a small number, but it's guaranteed to be at most 1 and at most ε/3, so it's small enough to work.In summary, to calculate the value of the limit lim (x->1) x2, we can use the ε-δ definition of the limit. By manipulating the expression |x2 - 1| < ε, we can determine that |x-1| < ε/3. Therefore, we can choose δ = min{1, ε/3} to satisfy the condition for the limit.
  • #1
Luscinia
17
0

Homework Statement


Calculate the value of the limit and justify your answer with the ε-δ definition of the limit.
lim (x->1) x2

Homework Equations


My professor gave us the hint that we have to take δ as 0<δ≤ k0 so that δ(ε)=min{k0,ε/ (k0+2)}

I'm guessing that k0 is meant to be any number though it's usually 1?

The Attempt at a Solution


I'm trying to relate |f(x)-1|<ε to |x-a|<δ
|x2-1| < ε
-ε < x2-1 <ε
-ε+1 < x2 <ε+1
(-ε+1)1/2-1 < x-1 < (ε+1)1/2-1

I'm not sure what to do from here. I'm guessing that
δ=(-ε+1)1/2-1 or (ε+1)1/2-1
but I have no clue where to go from there.

I've tried doing this another way as well to make use of a k0 from my professor's hint

|x2-1| < ε
-ε < x2-1 <ε
-ε+1 < x2 <ε+1
-ε+1 < x x < ε+1 where if we assume that |x|<k0=1, we can then assume that -ε+1 < k0x < ε+1
(-ε+1)/k0 - 1 < x-1< (ε+1)/k0 - 1
This still doesn't give me what my professor hinted at though. (I don't know what to do after that.)
Also, what does the min{_____,_______} mean?
 
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  • #2
Luscinia said:

Homework Statement


Calculate the value of the limit and justify your answer with the ε-δ definition of the limit.
lim (x->1) x2

Homework Equations


My professor gave us the hint that we have to take δ as 0<δ≤ k0 so that δ(ε)=min{k0,ε/ (k0+2)}

I'm guessing that k0 is meant to be any number though it's usually 1?

The Attempt at a Solution


I'm trying to relate |f(x)-1|<ε to |x-a|<δ
|x2-1| < ε
-ε < x2-1 <ε
-ε+1 < x2 <ε+1
(-ε+1)1/2-1 < x-1 < (ε+1)1/2-1

I'm not sure what to do from here. I'm guessing that
δ=(-ε+1)1/2-1 or (ε+1)1/2-1
but I have no clue where to go from there.

I've tried doing this another way as well to make use of a k0 from my professor's hint

|x2-1| < ε
-ε < x2-1 <ε
-ε+1 < x2 <ε+1
-ε+1 < x x < ε+1 where if we assume that |x|<k0=1, we can then assume that -ε+1 < k0x < ε+1
(-ε+1)/k0 - 1 < x-1< (ε+1)/k0 - 1
This still doesn't give me what my professor hinted at though. (I don't know what to do after that.)
Also, what does the min{_____,_______} mean?
k0 might be 1. It's just some unspecified number.

min{..., ...} means the minimum of the numbers in the list inside the braces.
 
  • #3
I think I managed to connect the dots together. I shouldn't have gotten rid of the -1 in |x2-1|. Instead, I should have just turned it into (x-1)(x+1) and move on from there.

Assuming that x+1<1 and δ<1,
-1<x-1<1 so if we add +2 to both sides, we get 1<x+1<3
|x+1||x-1|<3|x-1|<ε
|x-1|<ε/3

But if I assume that δ<1, then δ=min{1,ε/3} won't make sense?
 
  • #4
Luscinia said:
I think I managed to connect the dots together. I shouldn't have gotten rid of the -1 in |x2-1|. Instead, I should have just turned it into (x-1)(x+1) and move on from there.

Assuming that x+1<1 and δ<1,
Just assume that δ < 1. That makes |x - 1| < 1, which gives you bounds on |x + 1|, as you show below.
Luscinia said:
-1<x-1<1 so if we add +2 to both sides, we get 1<x+1<3
|x+1||x-1|<3|x-1|<ε
|x-1|<ε/3

But if I assume that δ<1, then δ=min{1,ε/3} won't make sense?
Now, choose δ = min{1, ε/3}. Typically, someone (else) would pick a small value for ε, so δ will typically be much smaller than 1.
 

1. How do you formally define a limit?

To formally define a limit, we use the notation lim x→a f(x) = L, which means that as x gets closer and closer to a, the value of f(x) gets closer and closer to L.

2. What are the conditions for proving a limit exists using formal definition?

There are three conditions that must be met in order to prove a limit exists using formal definition: the function must be defined at all points near the limit, the function must approach a specific value as x approaches the limit, and the value of the limit must be the same from both the left and right sides of the limit.

3. Can a limit exist even if the function is not defined at the limit point?

Yes, a limit can still exist even if the function is not defined at the limit point. This is because the definition of a limit only considers the behavior of the function as x approaches the limit, not the actual value of the function at the limit point.

4. How do you use the formal definition to prove a limit exists?

To prove a limit exists using formal definition, you must show that the limit expression is true for all positive values of ε, no matter how small. This involves finding a value for δ (delta) that satisfies the three conditions mentioned in question 2.

5. Are there any shortcuts for proving a limit exists?

No, there are no shortcuts for proving a limit exists using formal definition. This method is considered the most rigorous way to prove a limit exists, as it directly follows from the definition of a limit.

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