Question

Find limit of function of two variables (equation inside)

Answer 1

\[\lim_{(x,y) \rightarrow (1,0)} \frac{ xy - y }{ (x-1)^2 + y^2 }\]

Answer 2

WolframAlpha says that the limit doesn't exist, but I don't know how to prove that. My book suggests solving for (1,y) (x,x) (x,0) etc., but I get zero for all of those combinations. To me, it looks like the limit exists.

Answer 3

Define x and y as follows:
\[x=\sin \theta \ y=\cos \theta \]
now the limitation process would be
\[Lim_{x \rightarrow \pi/2} \frac{ \sin \theta \times \cos \theta - \cos \theta }{(\sin \theta - 1)^2 + \cos^2\theta } = Lim_{x \rightarrow \pi/2} \frac{ \cos \theta(\sin \theta -1) }{ 2(1 - \sin \theta) } = Lim_{x \rightarrow \pi/2} \frac{ - \cos \theta }{ 2 } = 0\]

Answer 4

@johnT So it tends to zero.

Answer 5

You just cant do that @Saeeddiscover
If you let x=sin(theta) and y=cos(theta)then, what you are assuming is that x^2+y^2=1 which is not given

Answer 6

let z=x-1 then the limit becomes