Limit of a multivariable function

• hnnhcmmngs
In summary, the conversation discusses how to solve the limit of (2x^2 + 3y^2)/(5xy) as (x,y) approaches (0,0). The person asking for help has tried using parametric and polar equations, but neither method worked. They are then advised to try taking limits along different paths and reminded that showing that the limit does not exist is often easier than showing that it does exist. The person asking for help is also encouraged to post their working so others can assist them better.
hnnhcmmngs

Homework Statement

If possible, calculate the following limit:
$\lim_{(x,y)\rightarrow (0,0)} {\frac{2x^2 + 3y^2}{5xy}}$

N/A

The Attempt at a Solution

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I tried using both parametric and polar equations to find the limit, but neither worked. Setting either x or y equal to zero also won't work because of the denominator. What method should I use to solve this?

Hello. The way that PhysicsForums works for homework problems: you provide what you have attempted, then we will guide you to a solution.
So what have you tried so far? Can you be more specific about the parametric and polar?

Like @scottdave said...
scottdave said:
The way that PhysicsForums works for homework problems: you provide what you have attempted, then we will guide you to a solution.

A technique that sometimes works is to take limits along various paths, such as along either axis or along a straight line through the origin or along various curves that pass through the origin. Finding the limit along various paths isn't enough to establish that a limit exists, but if you get different results along different paths, then you can say that the limit doesn't exist.

The fact that x and y occur to the same powers in both numerator and denominator makes things relatively easy in this problem.

Somewhat-strangely, it is usually easier to show that the limit does not exist than to show it existr -- and find the limit.Specially -so in 2D or higher.

scottdave
hnnhcmmngs said:
tried using ... polar equations
That should have solved it immediately. Please post your working. Maybe you did not understand what your equation was telling you.

So @hnnhcmmngs were you able to arrive at a conclusion on this limit?

What is the limit of a multivariable function?

The limit of a multivariable function is the value that the function approaches as the input variables approach a specific point. It can also be thought of as the value that the function "approaches" or "tends to" as the input variables get closer and closer to a certain point.

How is the limit of a multivariable function calculated?

The limit of a multivariable function is calculated by plugging in the values of the input variables into the function and observing the resulting output. As the input variables get closer and closer to the specific point, the resulting output values should also get closer and closer to a certain value, which is the limit.

Why is the limit of a multivariable function important?

The limit of a multivariable function is important because it helps us understand the behavior of the function at a specific point. It can also help us determine the continuity of a function and whether it has any discontinuities or holes at that point.

Can the limit of a multivariable function exist even if the function is not defined at that point?

Yes, the limit of a multivariable function can exist even if the function is not defined at that point. This is because the limit is based on the behavior of the function as the input variables approach the specific point, not the actual value of the function at that point.

How is the limit of a multivariable function affected by the number of input variables?

The limit of a multivariable function can be affected by the number of input variables, as it can become more complex to calculate and may involve multiple limits. However, the basic concept of approaching a specific point still applies, regardless of the number of input variables.

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