Trouble with delta epsilon proofs

In summary: And then they added them all up and saw that it would be less than epsilon. They probably had some intuition as to what choices would work based on previous experience, but it ultimately comes down to trial and error. In summary, the conversation discusses a proof of the limit theorem where the limit of a product of two functions is equal to the product of their limits. The book uses substitution and rearrangement to show that the limit is less than epsilon. The confusion lies in where the book got the expressions for delta, such as sqrt(ε/3) and ε/[3(1+|L1|)]. The book randomly chose these expressions to make each term in the sum less than epsilon/3, ultimately leading to the total sum being
  • #1
demonelite123
219
0
i know how to do basic proofs, but some proofs on the actual limit theorems confuse me. my textbook's choices for delta are very obscure and i have no idea how they even came up with them.

for the proof of the limit theorem where the limit of a product of 2 functions is equal to the product of their limits, my book did: f = L1 + (f-L1) and g = L2 + (g-L2). and they want to show that |f*g - L1*L2|< ε if 0<|x-a|<δ.

so with substitution and rearrangement they get |L1(g-L2)+L2(f-L1)+(f-L1)(g-L2)|< ε. since the limit of f as x approaches a is L1 and limit of g as x approaches a is L2, we can find positive numbers δ1, δ2, δ3, δ4 such that:

|f-L1|< sqrt(ε/3) if 0<|x-a|<δ1
|f-L1|< ε/[3(1+|L2|)] if 0<|x-a|<δ2
|g-L2|< sqrt(ε/3) if 0<|x-a|<δ3
|g-L2|< ε/[3(1+|L1|)] if 0<|x-a|<δ4

the remainder of the proof after the above step i understand but what confuses me is where and how did they get those expressions like sqrt(ε/3) and ε/[3(1+|L2|)]?
 
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  • #2
you're not supposed to fix delta at the start. suppose you want to find a delta such that expression1<delta implies expression2<epsilon:

start with:
expression1 < delta

then:
-manipulate inequalities-
-manipulate inequalities-
-manipulate inequalities-and suppose you end up with, for example,
expression2 < 3*delta^2 Then you just look at the final inequality and go, 'if i make delta=sqrt(ε/3)' then expression2 < ε.
 
  • #3
|L1(g-L2)+L2(f-L1)+(f-L1)(g-L2)|<=|L1(g-L2)|+|L2(f-L1)|+|(f-L1)(g-L2)|. They conveniently choose those deltas so that each term in the sum is less than epsilon/3. So the total sum is less that epsilon. For example, the first term is less than |L1|*(epsilon/(3(1+|L1|)). |L1|/(1+|L1|)<=1. So the product is less than epsilon/3. The sqrt(epsilon/3) inequalities come in handy for the last term.
 
Last edited:
  • #4
boboYO said:
you're not supposed to fix delta at the start. suppose you want to find a delta such that expression1<delta implies expression2<epsilon:

start with:
expression1 < delta

then:
-manipulate inequalities-
-manipulate inequalities-
-manipulate inequalities-


and suppose you end up with, for example,
expression2 < 3*delta^2


Then you just look at the final inequality and go, 'if i make delta=sqrt(ε/3)' then expression2 < ε.

how did the book get those 4 inequalities though? the expressions seem very random to me. sqrt(ε/3) and ε/[3(1+|L1|)] and ε/[3(1+|L2|)]. the book simply just wrote those 4 inequalities in the proof without showing how they got those expressions at all. that's what I'm confused about.
 
  • #5
demonelite123 said:
how did the book get those 4 inequalities though? the expressions seem very random to me. sqrt(ε/3) and ε/[3(1+|L1|)] and ε/[3(1+|L2|)]. the book simply just wrote those 4 inequalities in the proof without showing how they got those expressions at all. that's what I'm confused about.

They are random in the sense that they aren't the only choice you can make. Many choices will work. They are just one of them.
 
  • #6
Dick said:
|L1(g-L2)+L2(f-L1)+(f-L1)(g-L2)|<=|L1(g-L2)|+|L2(f-L1)|+|(f-L1)(g-L2)|. They conveniently choose those deltas so that each term in the sum is less that epsilon/3. So the total sum is less that epsilon. For example, the first term is less than |L1|*(epsilon/(3(1+|L1|)). |L1|/(1+|L1|)<=1. So the product is less than epsilon/3. The sqrt(epsilon/3) inequalities come in handy for the last term.

ok I understand how the 3 terms reduce down to ε/3 and add up together to ε in the end. so did they just kind of guess and check to see which deltas would work out like that? I followed the proof step by step from beginning to end and I couldn't figure out how they just produced those inequalities seemingly out of nowhere.
 
  • #7
oh ok I understand now. thank you both!
 
  • #8
demonelite123 said:
ok I understand how the 3 terms reduce down to ε/3 and add up together to ε in the end. so did they just kind of guess and check to see which deltas would work out like that? I followed the proof step by step from beginning to end and I couldn't figure out how they just produced those inequalities seemingly out of nowhere.

Yes, they just looked at the first term and said what's a delta choice that will make that less than epsilon/3. Same for the other terms.
 

Related to Trouble with delta epsilon proofs

1. What is a delta epsilon proof?

A delta epsilon proof is a mathematical technique used to prove the limit of a function. It involves using the concepts of delta (change in input) and epsilon (desired error) to show that for any given epsilon, there exists a delta such that the function's output is within epsilon of the limit.

2. Why are delta epsilon proofs important?

Delta epsilon proofs are important because they provide a rigorous and formal way to prove the limit of a function. They are essential in calculus and other areas of mathematics that deal with limits and continuity.

3. What are some common challenges when working with delta epsilon proofs?

Some common challenges when working with delta epsilon proofs include understanding the concept of limits, choosing the appropriate delta to match a given epsilon, and keeping track of multiple variables and inequalities.

4. How can I improve my skills in solving delta epsilon proofs?

Practicing and understanding the underlying concepts are key to improving skills in solving delta epsilon proofs. It may also be helpful to review and understand similar examples and seek help from a tutor or teacher if needed.

5. Are there any alternative methods to proving limits besides delta epsilon proofs?

Yes, there are other methods such as using the squeeze theorem or using algebraic manipulation to simplify the limit expression. However, delta epsilon proofs are the most commonly used and accepted method in mathematics.

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