Determining Limits for Multivariable Functions

[paul2211]

Homework Statement

$$\lim_{\substack{x\rightarrow 2\\y\rightarrow 2}} f(x,y)=\frac{2x^2+2xy+2x-xy^2-y^3-y^2}{2x^3-2x^2y+2x-x^2y^2+xy^3-y^2}$$

Homework Equations

N/A

The Attempt at a Solution

Well, I tried using the line y = 2, and let x approach 2 (as well as making x=2 and let y approach 2), and nothing really seems to cancel when I do that.

I also tried converting everything to polar form and let r approach 2$\sqrt{2}$, but again, everything just looks really messy.

This is a previous year's test question, and it's not worth much marks, so I think I'm missing some trick/procedure to simplify this question.

Also, I think this does not have a limit because the degree of the top (3) is less than the degree of the bottom (4); however, I'm not sure of how to prove this.

 

[SammyS]

What do you mean by "nothing really seems to cancel". You should get the indeterminate form, 0/0 .

Factoring the numerator gives $(x+y+1) (2 x-y^2)\,.$

Factoring the denominator gives $(x^2-x y+1) (2 x-y^2)\,.$

 

[paul2211]

When I said "nothing seems to cancel", I just meant that the variables don't cancel out. For example, we can sometimes sub in y=mx^2, into a function to balance out the powers and allow the x to divide out, leaving a limit that changes depending on m.

Anyway, thank you very much for helping me regarding this problem.