Hello I have an exercise to train to my final calculus 2 test that consists in finding the following limit:
$\lim_{(x,y)\rightarrow(1,0)}\frac{xy-y}{x^2+y^2-2x+1}$
If anyone could help me with this i would be grateful. Also, is there a specific strategy to doing multivariable limits?
PS: Sorry only found out now how to insert the latex thingy
On
The first thing I would do is change variables so the limit point is (0, 0): let y= x- 1, v= y. Then we have $\lim_{u\to 0, v\to 0}\frac{uv}{u^2+ v^2}$. Now, convert to polar coordinates. (u, v) will go to 0 along any path means that r goes to 0 for any $\theta$. The limit becomes $\lim_{r\to 0} \frac{r^2cos(\theta)sin(\theta)}{r^2}= cos(\theta)sin(\theta)$. The fact that that depends upon $\theta$ means that the original limit does not exist.
$\dfrac{y(x-1)}{(x-1)^2-1 +y^2+1}.$
Let $z:= x-1$, and consider
$\lim_{(z,y) \rightarrow (0,0)} |\dfrac{yz}{z^2+y^2}|$.
1)Let $y=t$; $z=t$; $t \rightarrow 0.$
Then : $\lim_{t \rightarrow 0} \dfrac{t^2}{t^2+t^2}=1/2.$
2) $y=t^2$; $z=t$.
Then
$\lim_{t \rightarrow 0}\dfrac{t^3}{t^4+t^2}=$
$\lim_{t \rightarrow 0}\dfrac{t}{1+t^2}=0.$
Limit does not exist.