How to solve a limit in two variables with an indeterminate form at (0,0)?

[Ray Vickson]

everything you are saying is correct, every single word. but now you are going against mark44, because he says that the polar form includes also paths that are not straight lines.

Not really: we are saying different things. When $r$ and $\theta$ are not linked we get straight lines for fixed $\theta$ and varying $r$. However, when $r = R(\theta)$ or $\theta =\Theta(r)$ we are just looking at the polar equation for some curve, and in that case anything goes. Mark44 was talking about varying both $r$ and $\theta$, but I was talking about something different.

 

[Kenneth1997]

what you are saying is what i have been saying since my first comment. mark is saying that converting to the polar form we are talking about considers also NON linear paths (i never said that the polar form didnt consider linear paths as you implied), which is the opposite as what you have stated. what would even be the point of finding a polar form such as r(theta)?

 

[Kenneth1997]

it wouldn't even be possible for the function i was trying to solve, and certainly it wouldn't help for finding a limit

 

[Mark44]

what you are saying is what i have been saying since my first comment.

No, it is not. You said this in post #4.

i think what you are saying is incorrect, the polar form doesn't include non linear paths, so you can't be sure if the limits exist.

mark is saying that converting to the polar form we are talking about considers also NON linear paths

Yes, and both Ray Vickson and I agree on this, if you read closely what he has written. You have at least one misconception that seems to be keeping you from understanding, and that is, if $\theta$ doesn't appear in a polar equation, $\theta$ is necessarily constant. That is NOT true.
Simple example: Consider the equation x = 1 in the Cartesian plane. y doesn't appear in this equation, so it can take on any real value. The graph of x = 1 is a vertical line through the point (1, 0). Every point on this line has an x-coordinate of 1, but the y-coordinate is completely arbitrary.

 

[epenguin]

Have you tried just two simple paths, a horizontal one and a vertical one? I.e. The limit as x→0 when y = 0. And if it has one then the other way around. Do they exist? And if they do are they the same? Maybe this would not always be conclusive, but seems to me it is here and is not difficult.

 

[berkeman]

Thread closed for Moderation...

After a Mentor discussion and some thread cleanup, the thread will remain closed. Thanks to everyone who contributed to the thread.