What is the Limit of polar coordinates?

In summary, the conversation discusses the process of evaluating a limit by converting to polar coordinates. The solution involves simplifying the equation and considering different values of theta, leading to the conclusion that the limit does not exist.
  • #1
andrassy
45
0

Homework Statement

I need to evaluate this limit by converting to polar coordinates:

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



Homework Equations

x = rcos(theta), y = rsin(theta)



The Attempt at a Solution

So switching to polar I get:

[(rcos(theta))^2 + rcos(theta)rsin(theta) + (rsin(theta))^2] / (rcos(theta))^2 + (rsin(theta))^2

By pulling out the r^2 from the the top of the equation and the bottom of the equation, they can cancel. Then the denominator is cos(theta)^2 + sin(theta)^2 which equals 1.

So we get the limit of cos(theta)^2 + cos(theta)sin(theta) + sin(theta)^2 but I don't know what to do from here because this is the limit as r goes to 0 and there is no r?

I'm kinda stuck here...what can I do? We didn't really get taught this so I could be missing something simple.

Thanks!
 
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  • #2
You can simplify a little more down to 1+cos(theta)sin(theta). There is always the possible answer that the limit doesn't exist, right? What do you say in this case and why?
 
  • #3
Dick said:
You can simplify a little more down to 1+cos(theta)sin(theta). There is always the possible answer that the limit doesn't exist, right? What do you say in this case and why?
Right I figured the limit did not exist. Does it have to do wiht the cos(theta)sin(theta)? So As r goes to 0, the function is just 1cos(theta)sin(theta) for whatever value of theta which will oscillate. Is that the correct way of thinking about it?
 
  • #4
Look at it this way. For the limit to exist the limit has to be independent of the way (x,y) approaches (0,0). If you set y=0, and let x->0, what's the limit. (This is the theta=0 case, right? Check it in your polar formula.) Now set y=x and let x->0. (This is the theta=pi/4 case. Try putting that into your polar formula as well.). So right, the limit does not exist. Because it depends on theta.
 

What is the limit of polar coordinates?

The limit of polar coordinates refers to the value that a function approaches as the input values approach infinity or a specific point in the polar coordinate plane.

How is the limit of polar coordinates calculated?

The limit of polar coordinates is calculated by converting the polar coordinates to Cartesian coordinates and then applying the limit rules for Cartesian coordinates, such as substitution and factoring.

What happens if the limit of polar coordinates does not exist?

If the limit of polar coordinates does not exist, it means that the function does not approach a specific value as the input values approach infinity or a specific point in the polar coordinate plane. This could be due to oscillations or undefined behavior of the function.

Can the limit of polar coordinates be different from the limit of Cartesian coordinates?

Yes, the limit of polar coordinates can be different from the limit of Cartesian coordinates because the two coordinate systems represent different ways of describing a point in a plane. While Cartesian coordinates use the x and y axes, polar coordinates use the distance from the origin and the angle from the positive x-axis.

What is the significance of finding the limit of polar coordinates in scientific research?

Finding the limit of polar coordinates is important in scientific research as it can help understand the behavior of functions in polar coordinates, which are commonly used in fields such as physics and engineering. It can also aid in solving complex equations and predicting the behavior of systems in real-world applications.

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