SOLUTION: lim (x, y) -> (0, 0) {{{ (xy + y^3) / (x^2 + y^2) }}} I am having a hard time knowing how to solve this properly. Simply plugging in the values doesnt work. Neither does fact

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: lim (x, y) -> (0, 0) {{{ (xy + y^3) / (x^2 + y^2) }}} I am having a hard time knowing how to solve this properly. Simply plugging in the values doesnt work. Neither does fact      Log On


   

Question 1039497: lim (x, y) -> (0, 0) +%28xy+%2B+y%5E3%29+%2F+%28x%5E2+%2B+y%5E2%29+
I am having a hard time knowing how to solve this properly.
Simply plugging in the values doesnt work.
Neither does factoring.
Which lead me to try y=0 [then solve], and x=0 [then solve].
if +y+=+0+, then I end up with +0+%2F+x%5E2+
if +x+=+0+, then I end up with +y+
Because the two answers dont match up, I am led to believe that the limit does not exist (DNE).

----
Then, as I was looking through my notes I noticed another method that my teacher suggested we try. It involves transforming the polar coordinates (replacing +x+ with +r+cos+theta+ and replacing +y+ with +r+sin+theta+
Upon doing this I ended up with:
lim r-> 0
r (cos sin theta) + (sin^3 theta) = 0
So does the limit exist? Or is it zero?
Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!



The limit may very well not exist.  Always test for this
possibility first.

Let's test for this:

If the value of the limit is not the same for all possible 
approaches, or paths, to (0,0), then the limit does not exist.

Let's choose the path y=kx for some constant k.
So we substitute kx for y:

%22%22=%22%22

%22%22=%22%22 

%22%22=%22%22

%22%22=%22%22

%22%22=%22%22

When x approaches 0, this approaches

k+%2F+%281+%2B+k%5E2%29  

which has different values for different choices of k.
For instance, it's 0 if k=0 and 1/2 if k=1, and 2/5 if k=2, etc.

That means the limit cannot exist, for it must approach
the same value regardless of the path taken to the origin.
But as we see this is not the case.

Answer:  the limit does not exist.

Edwin