Why is it possible to calculate multivariable limits using polar coordinates? Why is it possible to calculate multivariable limits using polar coordinates? Let's say I'm looking for some $\lim_{(x,y) \to (0,0)}$ and I'm substituting $x = r cos\theta$ and $y = rsin\theta$ so that I can look at $\lim_{r \to 0}$.
Why can I do this? Am I not just looking at "straight lines" going to $(0,0)$ now? What about all the other possible sequences that converging in straight lines to (0,0)?
 A: While $r$ is going to $0$, $\theta$ is arbitrary. So, $\theta$ can freely change however it wants, as long as the radius is going to zero (that is, the convergence is uniform in $\theta$).
EDIT: See the following link for rigorous details: Polar coordinates for the evaluating limits
A: "Why is it possible to calculate multivariable limits using polar coordinates? Let's say I'm looking for some 
lim
(x,y)→(0,0)
and I'm substituting  x=rcosθ and y=rsinθ so that I can look at 
lim
r→0. Why can I do this? Am I not just looking at "straight lines" going to 
(0,0) now? What about all the other possible sequences that converging in straight lines to (0,0)?"
Yes, doing it that way is wrong!  But if you show that the function goes to 0 as r goes to 0 without any reference to $\theta$, you are not taking the limit along any specific line.  You are just saying that, for a point, (x, y), close enough to (0,0) (and in polar coordinates, the distance to (0, 0) is measured by r alone) the function is close enough to the limit.
A: You are sending $x$ and $y$ to zero. This is satisfied by sending $r$ to zero because both $x$ and $y$ are proportional to $r$ in polar coordinates. Regardless of $\theta$, $\cos(\theta)$ and $\sin(\theta)$ are finite numbers, and thus as $r\rightarrow 0$, both $x=r\cos(\theta)\rightarrow0$ and $y=r\sin(\theta)\rightarrow0$. In this case you've turned a multivariable limit into a single variable limit, which is incredibly valuable. 
