• stvoffutt
In summary, we are trying to prove the limit of x^n*sin(1/x) as x approaches 0 for all n belonging to natural numbers. To do this, we can use the definition of a limit and induction. We start by showing that the limit is true for n=1, and then assume it is true for some n and try to prove it for n+1. We can also use the fact that -1<=sin(1/x)<=1 and make the substitution x=1/t to help determine which L to use in our proof.
stvoffutt

Homework Statement

Using the definition of |x-a|<delta implies |f(x) - L|<epsilon, prove that lim x->0 x^n*sin(1/x) holds for all n belonging to natural numbers.

Homework Equations

Definition of a limit

The Attempt at a Solution

Ok, so when I see "prove for all n belonging to natural numbers" I immediately think induction. So this is what I have done so far. For n=1 lim x->0 x^n*sin(1/x) is true; the limit is zero. Now I will assume lim x->0 x^n*sin(1/x) for some n is true, then I need to show that n+1 is also true. So I start using the definition of the limit and don't know what my L should be and how to use induction along with this definition of a limit. Please help and thank you in advanced.

Last edited:
$$\mbox{If you can prove that }\lim_{x\to 0}x^{n}sin\left(\frac{1}{x}\right)\mbox{ is bounded by two functions whose limit is L, then you've proven that the limit exists.}$$
$$\mbox{(You don't have to invoke the Squeeze Theorem.)}$$

$$\mbox{Start with }-1\leq \sin\left(\frac{1}{x}\right)\leq 1\mbox{. What can you say about }x^{n}sin\left(\frac{1}{x}\right)\mbox{ ?}$$

$$\mbox{It isn't completely necessary, but it might help you see what L to use if you make the substitution x=1/t, then }$$

$$\lim_{x\to 0}x^{n}sin\left(\frac{1}{x}\right) = \lim_{t\to\infty}\left(\frac{1}{t}\right)^{n}sin\left(t\right)$$

1. What is a Delta Epsilon proof?

A Delta Epsilon proof is a method used in advanced calculus and mathematical analysis to formally prove the limit of a function. It involves using the formal definitions of limits, specifically the delta-epsilon definition, to show that for any given value of epsilon, there exists a corresponding value of delta that satisfies the definition of a limit.

2. How is a Delta Epsilon proof different from other methods of proving limits?

A Delta Epsilon proof is considered to be one of the most rigorous and formal methods of proving limits. It involves directly using the definition of a limit to show that the limit exists and is equal to a specific value. Other methods, such as the squeeze theorem or the use of L'Hopital's rule, rely on more advanced concepts and may not be applicable to all functions.

3. What are the key components of a Delta Epsilon proof?

The key components of a Delta Epsilon proof include the formal definition of a limit, which states that for every epsilon greater than zero, there exists a corresponding delta such that the absolute value of the difference between the input value and the limit is less than epsilon. The proof also involves using algebraic manipulations and logical reasoning to show that the limit is indeed equal to the desired value.

4. When is it necessary to use a Delta Epsilon proof?

A Delta Epsilon proof is necessary when proving the limit of a function that is not continuous or does not have an obvious limit, such as functions with jump discontinuities or non-removable discontinuities. It is also commonly used in more advanced mathematical analysis to prove the convergence of series and sequences.

5. What are some tips for successfully completing a Delta Epsilon proof?

Some tips for successfully completing a Delta Epsilon proof include carefully defining the limit and understanding the concept of epsilon and delta. It is also important to use algebraic manipulations and logical reasoning to simplify the problem and arrive at the desired conclusion. Additionally, practice and familiarity with the concept and its applications can greatly improve the success rate of completing a Delta Epsilon proof.

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