Hi everyone
I need to find whether this function is continues at (1,2).
Well actually the question is how do I find the limit as (x,y) goes to (1,2).
how do I do that?
Thank you
In order for f to be continuous, $\displaystyle \lim _{(x,y) \to (1,2)}$ must be equal to $\displaystyle 0$.
First I notice that $\displaystyle \frac{\partial f}{\partial x}(1,2)$ and $\displaystyle \frac{\partial f}{\partial y}(1,2)$ doesn't seem to be easy to calculate. Hence we must use the definition of limit to prove what they ask for.
I'm not sure about how to proceed, but you have to reach $\displaystyle f(1,2)$ via a line and if it gives $\displaystyle 0$ then try with a quadratic function. If it still gives $\displaystyle 0$, then you can suppose that the limit exists and is worth $\displaystyle 0$, hence the function to be continuous at $\displaystyle (1,2)$. From it you would have to show that $\displaystyle \forall \varepsilon >0$, $\displaystyle \exists \delta(\varepsilon)$ such that if $\displaystyle |(x,y)-(1,2)|<\delta(\varepsilon)$, then $\displaystyle |f(x,y)-f(1,2)| < \varepsilon$.
I'll wait for further help as well.
Yes I know. But this is not possible I believe. That's why you suppose it's true and try to prove it by definition. If you aren't able to prove it by any mean, then try another curve.
Edit: By the way, to show that it holds for any curve of approaching is precisely proving by definition that the limit exists!
Anyway I've plotted it and it didn't seem to be continuous. (I might be wrong of course). So you would have to try to approach the limit with a curve $\displaystyle y=ax+b$ and then $\displaystyle y=ax^2+bx+c$. Hopefully it will be sufficient to prove that the limit doesn't exist.
Another edit : This is a hard one! I've tried to approach (1,2) by a curve y=ax, y=x^2, x=ay and it's all worth 0 if the constant a satisfy some equalities. Hence I've no idea how to show the limit doesn't exist.
I'd be glad if anyone could help us
Oh... thanks for the help. I almost get the result.
Here's my work: Let $\displaystyle u=(x-1)$, $\displaystyle v=(y-2)$. So the limit we're looking for (call it $\displaystyle I$) becomes equal to $\displaystyle \lim_{(u,v)\to (0,0)} \frac{uv^2}{u^2+\sin ^2(v)}$.
Hence $\displaystyle I=\lim_{(u,v)\to (0,0)}\frac{|uv||v|}{u^2+\sin ^2 (v)} \leq \lim_{(u,v)\to (0,0)} \frac{(u^2+v^2)|v|}{2(u^2+ \sin ^2(v))}=\lim_{(u,v)\to (0,0)} \frac{|v|}{2}=0$. As I $\displaystyle \geq 0$, I must equal $\displaystyle 0$. Hmm this is only true if $\displaystyle u$ approaches $\displaystyle 0$ from positive values.... So we're not yet done. I hope someone will finish it.