Thread: help with limits using two variables

1. help with limits using two variables

How do I find the limit of

lim
(x,y)->(0,0)

( x - xy + 3 ) / ( (x^2)y + 5xy - y^3 )

I know I have to use

lim
h->0
( f(0,h) - f(0,0) ) / h

but I get 3/0

2. Re: help with limits using two variables

You have tried only the direction parallel to the y-axis. There are infinitely many pathways to approach (0,0).

Take a shot at the x-direction. Get really creative and try the direction defined by x = y.

I thought it was fun to convert it to polar coordinates and then think about 'r' as it approaches zero. You can pick an angle you like for any approach.

What makes you think your result is inappropriate?

3. Re: help with limits using two variables

Originally Posted by TKHunny
You have tried only the direction parallel to the y-axis. There are infinitely many pathways to approach (0,0).

Take a shot at the x-direction. Get really creative and try the direction defined by x = y.

I thought it was fun to convert it to polar coordinates and then think about 'r' as it approaches zero. You can pick an angle you like for any approach.

What makes you think your result is inappropriate?
It's important to note that for the limit to exist, it needs to approach the same value from EVERY possible path...

4. Re: help with limits using two variables

So you are saying to try plugging x equals y?, but how does that help solving the limit?

5. Re: help with limits using two variables

Originally Posted by Sneaky
So you are saying to try plugging x equals y?, but how does that help solving the limit?
Like I said, the point is that the limit only exists if it approaches the same value from EVERY possible path. As soon as you can show that the function approaches a different value from a different path, that's enough to show that the limit does not exist...

6. Re: help with limits using two variables

I'm suggesting only interesting exploration. x = y is only one specific direction of approach.