# Solving Multivariable Limit: Find Limit or Show No Existence

• Laura1321412
In summary, the problem asks to find if the limit exists or not for the given function (x^2 * sin^2y)/(x^2 + 2y^2) as (x,y) approaches (0,0). The attempt at a solution involves using the limit definition and checking the conditions |sin^2y| <= 1 and |x^2sin^2y/x^2 + 2y^2| <= 1. However, the last condition of plugging in x = y and y = x needs to be checked as well to prove that the limit exists.
Laura1321412

## Homework Statement

Find that the limit exists, or show that the limit does not exist.

lim (x,y) --> (0,0) of (x^2 * sin^2y)/(x^2 + 2y^2)

??

## The Attempt at a Solution

i used the lines x=0 y=0 and both times got the limit 0 so i attempted to prove the limit exists

I attempted to use the limit definition,

1 >= |sin^2| >= 0
x^2 + 2y^2 >= x^2
x^2+2y^2 >= x^2sin^2y
|x^2sin^2y/x^2+2y^2| <= 1

uhmm... I am not sure what do after this or really what the heck I am doing. HELP? :)

You are forgetting to check one last condition before proving that the limit exist! Not sure if it would do anything but try plugging x = y and y = x. If this checks out ( = 0) then you can be sure that the limit exist and try to prove it.

## 1. What is a multivariable limit?

A multivariable limit, also known as a limit of a multivariable function, is a mathematical concept that describes the behavior of a function as its input values approach a specific point in a multi-dimensional space.

## 2. How do you solve a multivariable limit?

To solve a multivariable limit, you can use several methods such as substitution, algebraic manipulation, or taking limits along specific paths. It is important to check for the existence of a limit before attempting to solve it.

## 3. What does it mean when a multivariable limit does not exist?

If a multivariable limit does not exist, it means that the function does not approach a single value as its inputs approach a specific point. This could be due to the function having different values along different paths or approaching different values from different directions.

## 4. How can you show that a multivariable limit does not exist?

To show that a multivariable limit does not exist, you can use different methods such as the epsilon-delta definition, squeeze theorem, or comparison test. These methods involve finding two different paths that approach the point in question and show that the function has different values along these paths.

## 5. Why are multivariable limits important?

Multivariable limits are important because they help us understand the behavior of a function in a multi-dimensional space. They are also used in many applications, such as optimization problems in engineering and physics, to determine the maximum or minimum values of a function.

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