# Continuity & Limits

• Oct 6th 2008, 03:37 PM
Len
Continuity & Limits
For what points is the function continuous?

$\displaystyle f(x,y)=\frac{xy}{x^2-y}$

Intuition tells me that it would be continuous everywhere except where the denominator equals zero so... Would it be continuous everywhere except on the parabola $\displaystyle y=x^2$?

Also

$\displaystyle f(x,y)=\frac{x^4-4y^4}{x^2+2y^2}$

Find the limit of $\displaystyle f(x,y)$ as $\displaystyle (x,y)\rightarrow(0,0)$

Thanks for the help.(Happy)
• Oct 6th 2008, 03:42 PM
icemanfan
Quote:

For what points is the function continuous?

$\displaystyle f(x,y)=\frac{xy}{x^2-y}$

Intuition tells me that it would be continuous everywhere except where the denominator equals zero so... Would it be continuous everywhere except on the parabola $\displaystyle y=x^2$?
Yes.

Quote:

$\displaystyle f(x,y)=\frac{x^4-4y^4}{x^2+2y^2}$

Find the limit of $\displaystyle f(x,y)$ as $\displaystyle (x,y)\rightarrow(0,0)$
$\displaystyle f(x, y) = \frac{(x^2 - 2y^2)(x^2 + 2y^2)}{(x^2 + 2y^2)}$.

Does this help?
• Oct 6th 2008, 03:44 PM
Len
Quote:

Originally Posted by icemanfan
Yes.

$\displaystyle f(x, y) = \frac{(x^2 - 2y^2)(x^2 + 2y^2)}{(x^2 + 2y^2)}$.

Does this help?

Saw that after I typed it up, but thought I better clarify :P, thanks a lot :D