# Proving a limit using epsilon/delta

• ImAnEngineer
In summary, to prove that the limit is equal to 0 using the epsilon-delta method, we can use the fact that yz<=1/2*(y^2+z^2). By rewriting the expression and substituting in this inequality, we can find a relationship between epsilon and delta and complete the proof.
ImAnEngineer

## Homework Statement

Prove that $$\mathop {\lim }\limits_{(x,y,z) \to (0,0,0) } \frac{xyz}{{x^2+y^2+z^2 }} = 0$$
using the epsilon-delta method

## The Attempt at a Solution

$$0<|(x,y,z)-(0,0,0)|=\sqrt{x^2+y^2+z^2}<\delta$$
Now I have to rewrite:
$$0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|<\epsilon$$
So that I find a relationship between epsilon and delta.

This is where I get stuck... I can't figure out how to do that.

This is one of my attempts:
$$0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|\leq \left|\frac{xyz}{{x^2}}\right|=\left|\frac{yz}{{x}}\right|$$

Any help is very much appreciated!

Last edited:
Hint: yz<=1/2*(y^2+z^2)

Then it is easy:

$$0<\left|\frac{xyz}{x^2+y^2+z^2}-0\right|\leq\left|\frac{xyz}{y^2+z^2}\right|\leq\left|\frac{xyz}{2yz}\right|=\left|\frac{x}{2}\right|<\frac{\delta}{2}=\epsilon$$

How did you come up with: yz<=1/2*(y^2+z^2) ?
(Why is it true at all?)

## What is the definition of a limit using epsilon/delta?

The definition of a limit using epsilon/delta states that for a function f(x) and a given value L, the limit of f(x) as x approaches a is L if for any positive real number ε, there exists a positive real number δ such that |f(x) - L| < ε whenever 0 < |x - a| < δ.

## How do you prove a limit using epsilon/delta?

To prove a limit using epsilon/delta, you must begin by stating the definition of the limit. Then, you must use algebraic manipulation and logical reasoning to show that for any given ε, you can find a corresponding δ that satisfies the limit definition. This is typically done by setting an expression for δ in terms of ε and showing that it works for any ε > 0.

## What is the importance of using epsilon/delta in proving limits?

Epsilon/delta proofs are important because they provide a rigorous and precise way to prove limits, which is essential in mathematical analysis and other fields that rely on limits. They also help to illustrate the concept of a limit and provide a framework for understanding the behavior of a function near a specific point.

## What are some common mistakes when proving limits using epsilon/delta?

Some common mistakes when proving limits using epsilon/delta include not stating the definition of the limit at the beginning, not using algebraic manipulation correctly, and not considering all possible values of ε and δ. It is also important to be careful with inequalities and to clearly explain each step of the proof.

## Can a limit be proven using epsilon/delta if it does not exist?

No, a limit cannot be proven using epsilon/delta if it does not exist. The definition of a limit using epsilon/delta requires that the function approaches a specific value L as x approaches a. If the limit does not exist, then there is no single value that the function approaches, making it impossible to satisfy the limit definition for all ε and δ values.

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