# Multivariable Calculus - a question of limits

• PhysicalProof
In summary, the conversation is about proving a statement using the meaning of a limit and the given equations epsilon>0 and delta>0. The attempt at a solution involves elaborating on the inequality in (1) and using polar coordinates to simplify the equation, but the limit definition must be used. A hint is given to use the fact that |x|<r and |y|<r, where r=sqrt(x^2+y^2).
PhysicalProof

## Homework Statement

Prove the following, using the meaning of a limit:

## Homework Equations

epsilon > 0, delta > 0

0 < sqrt(x^2 + y^2) < delta
| f(x) - 0 | < epsilon (1)

## The Attempt at a Solution

So, I know that I have to elaborate on the inequality in (1), further. However, I'm not sure of how to go about this. If I start transposing it seems to get much too messy and I end up with a proposed delta which seems far too complex. Is there some simplification that I'm overlooking here?

Any help is really appreciated, ta.

Could you not use polar coordinates instead?
Then the equation would be much simpler and the limit is just as r goes to 0

I did consider that but I'm pretty sure the limit definition must be used here (as a requirement).

Does anyone have any ideas of how to simplify this beast?

EDIT: Really need this one clarified guys. Does anyone have any hints at all? Even a vague idea.

Hint: use the fact that |x|<r and |y|<r, where r=sqrt(x^2+y^2).

vela said:
Hint: use the fact that |x|<r and |y|<r, where r=sqrt(x^2+y^2).

Bingo, great hint, thank you! Subtle yet it serves me well :D♦

Last edited:

## 1. What is multivariable calculus?

Multivariable calculus is a branch of mathematics that deals with the study of functions of multiple variables. It involves the study of limits, derivatives, and integrals of functions with more than one independent variable.

## 2. Why is multivariable calculus important?

Multivariable calculus is important because it provides tools for analyzing and solving real-world problems involving multiple variables. It is used in many fields such as physics, engineering, economics, and statistics.

## 3. What are the key concepts in multivariable calculus?

The key concepts in multivariable calculus include partial derivatives, chain rule, directional derivatives, gradient, double and triple integrals, and optimization. These concepts are used to understand and solve problems involving functions of multiple variables.

## 4. What is the role of limits in multivariable calculus?

Limits are used in multivariable calculus to define the behavior of a function as the independent variables approach a certain value. They are essential in understanding the continuity and differentiability of functions of multiple variables.

## 5. How is multivariable calculus different from single variable calculus?

Multivariable calculus deals with functions of multiple variables, whereas single variable calculus deals with functions of only one variable. This means that the concepts and techniques used in multivariable calculus are more complex and involve more variables compared to single variable calculus.

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