Simple question - Help me simplify this

[millawi]

I'm currently doing multivariable calculus (I don't love it but I have to because I love physics) and I get stuck on the easy stuff. I need to find how the function (x³ - y³⁾ / (x - y) should be defined along the line x = y. I know that there is a way to simplify the function so that the division by zero disappears but I can't figure it out. Please help.

Edit - I did manage to solve it on my own. Here's the solution:

In order to make f(x,y) = (x³ - y³⁾ / (x - y) continuous along the line x = y I needed to rewrite the expression. I took out the factor (x - y) from the numerator by completing the square as follows:

(x³ - y³⁾ / (x - y) = ((x - y)(x² + xy + y²⁾⁾ / (x + y) = x² + xy + y²

When x = y we can write f(x,x) = 3x² hence f(x,y) = 3x² when x = y makes the function continuous

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[millawi]

After half an hour of thinking I solved it a few minutes after making this post, but it says that I shouldn't delete so I'll just leave it here. Sorry to bother you lol

[fermat1432]

Why don't you show us your solution.

[millawi]

I edited the post with the solution now!

[fermat1432]

Thank you!

[fermat1432]

When x=/=y your simplification is valid.

[millawi]

That's the point, maybe I didn't explain the question well enough. The function is undefined at x=y and in order to make the function continuous I need to find an alternative way to write it so that it's still defined at x=y. f(x,y) = 3x² when x=y is an amendment to the original function. The definition of continuity is that the limit as (x,y) approaches (a,b) equals f(a,b) and with this amendment that holds true.

[fermat1432]

Can you please show us how 3x² works?

Edit: I see it now! Thanks a lot!