Limit problem

[edumat]

I need to prove that this limit doesn't exists:

\(\displaystyle lim_{(x,y) -> (1,2)}\,\dfrac{xy-2x-y+2}{x^2+y^2-2x-4y+5}\)

With the two-path test, all I get is 0, I could not find a different result.

Paths I've already tried:
x=1
y=2
y=2x
y=2x²
y=-x+3

 

[DrPhil]

I need to prove that this limit doesn't exists:

\(\displaystyle lim_{(x,y) -> (1,2)}\,\dfrac{xy-2x-y+2}{x^2-y^2-2x-4y+5}\)

With the two-path test, all I get is 0, I could not find a different result.

Paths I've already tried:
x=1
y=2
y=2x
y=2x²
y=-x+3

Please check the signs of all terms in the denominator. As typed, the limit really is zero!

BTW, use "\dfrac" in place of "\frac" in LaTeX to get a larger font.

EDIT: I suspect the denominator should have +y^2 instead of -y^2.

 

[edumat]

Oh, sorry, it was +y^2

Thanks for noticing and the tip

 

[HallsofIvy]

With x= 1, y= 2, \(\displaystyle \dfrac{xy- 2x- y+ 2}{x^2- y^2- 2x- 4+ 5}= \dfrac{1(2)- 2(1)- 2+ 2}{1^2- 2^2- 2(1)- 4+ 5}= \dfrac{0}{-7}= 0\) so the limit exists and is, in fact, 0. The only time there would be any difficulty would be if both numerator and denominator were 0. Of course, if the denominator were 0 and the numerator were not, then we could immediately say that the limit did not exist.

 

[DrPhil]

Oh, sorry, it was +y^2

Thanks for noticing and the tip

When choosing paths for evaluation, does it help to factor the numerator and denominator?

\(\displaystyle \dfrac{xy- 2x- y+ 2}{x^2+ y^2- 2x- 4y+ 5}= \dfrac{(x-1)(y-2)}{(x-1)^2 + (y-2)^2}\)