Delta Epsilon Limit Proof

In summary, the problem asks to prove that if the limit of g(x) as x approaches infinity is infinity and g(x) is less than or equal to f(x) as x approaches some value a, then the limit of f(x) as x approaches a is also infinity. This can be proven using the formal definition of limits, by starting with the δ-ε definition and showing the necessary steps to arrive at the desired conclusion.
  • #1
Jimbo57
96
0

Homework Statement



Prove, using the formal definition of limits:

If lim (x->inf) g(x) = inf and g(x) leq f(x) for x->a, then lim (x->a) f(x)=inf.

leq = less than or equal to.

Homework Equations


The Attempt at a Solution



Honestly, I'm not even sure where to start on this one. Anyone bored enough to show how they would solve it?
 
Physics news on Phys.org
  • #2
Start with the δ-ε definition of a limit.

Show some work so we can help you. That's a rule for this Forum.

BTW: Don't you mean lim (x → a) g(x) = ∞ , NOT lim (x → ∞) ?
 

1. What is a Delta Epsilon Limit Proof?

A Delta Epsilon Limit Proof is a mathematical method used to formally prove the limit of a function. It involves using the concept of delta and epsilon to show that as the input to a function approaches a specific value, the output of the function approaches a certain value.

2. Why is a Delta Epsilon Limit Proof important?

Delta Epsilon Limit Proofs are important because they provide a rigorous and precise way to prove the existence of limits for functions. They are also used in many areas of mathematics and science, such as calculus and analysis, to prove theorems and solve problems.

3. What is the difference between delta and epsilon in a Delta Epsilon Limit Proof?

In a Delta Epsilon Limit Proof, delta represents the distance between the input value and the limit point, while epsilon represents the distance between the output value and the limit value. Delta is usually chosen to be a small positive number, while epsilon can be any positive number.

4. What are the steps involved in a Delta Epsilon Limit Proof?

The steps involved in a Delta Epsilon Limit Proof are as follows:

1. Start by assuming that the limit of the function exists.

2. Choose a value for epsilon that represents the desired level of accuracy.

3. Use algebraic manipulations to find a corresponding value for delta that satisfies the definition of the limit.

4. Prove that for any input value within delta units of the limit point, the corresponding output value is within epsilon units of the limit value.

5. Are there any limitations to using Delta Epsilon Limit Proofs?

While Delta Epsilon Limit Proofs are a powerful tool for proving limits, they may not always be practical or feasible to use. In some cases, alternative methods may be more efficient. Additionally, Delta Epsilon Limit Proofs may not work for all types of functions, such as discontinuous or non-differentiable functions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
791
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
946
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
19
Views
1K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
782
Back
Top