Examine $\lim_{(x,y)\to (0,0)}\frac{ax^2+bxy+cy^2}{x^2+y^2}=0$ The question is what can we conclude about $a,b,c$ if $\displaystyle \lim_{(x,y)\to (0,0)}\frac{ax^2+bxy+cy^2}{x^2+y^2}=0$. Let $f(x,y)$ be the given function.
Firstly, if we switch to polar coordinates we get that $f(x,y)=f(\theta)=a\cos^2\theta+b\cos\theta \sin\theta+c\sin^2\theta$. I can't see how this can give us more information.
We can try this approach: the limit exists and it is equal to $0$, so if we approach $(0,0)$ from any path, the function will take the value $0$. For example, we can examine the paths
$y=x$ then $\displaystyle f(x,x)=\frac{ax^2+bx^2+cx^2}{x^2+x^2}=\frac{a+b+c}{2}$ is $x$ independent so it must be equal to $0\Leftrightarrow  a+b+c=0$. I tried some function examples when $a,b,c$ follow that condition in Wolfram Alpha but their limit didn't exist when $(x,y)\to(0,0)$. Therefore $a+b+c=0$ doesn't seem to be a sufficient condition.
another path: $y=0$ then $\displaystyle f(x,0)=\frac{ax^2}{x^2}=a$ it must be equal to $0$.
Similarly the path $x=0$ gives us $\displaystyle f(0,y)=\frac{cy^2}{y^2}=c=0$ and by combining the results we get that $a=b=c=0$. 
This conclusion makes me think that I've done a mistake somewhere.
Any hints?
Thank you in advance!

Consider a linear approach $y=mx$. Since the multivariable limit exists, we find that 
$$
\lim_{x \to 0} \frac{ax^2+bmx^2 + cm^2x^2}{x^2+m^2x^2} = \frac{a + bm+cm^2}{1+m^2}
$$
exists and does not depend on $m$. However, obviously this limit will depend on $m$ if we don't impose some conditions on $a,b,c$.
Taking $m=0$ the limit must equal $a$ and taking $m\to \infty$ the limit must equal $c$, so we must have $a=c$. Now the limit reduces to
$$
\frac{a(1+m^2) + bm}{1+m^2} = a + \frac{b m}{1+m^2}
$$
which must equal $a$, so we find $b=0$.
Thus we have shown that in order for the multivariable limit to exist, we must have $a=c$ and $b=0$. Moreover, for any $a$
$$
\lim_{(x,y)\to(0,0)}\frac{ax^2 + 0xy + ay^2}{x^2+y^2} = \lim_{(x,y) \to (0,0)} a = a
$$
which does exist. Thus, we have show that the limit exists if and only if $a=c$ and $b=0$, and that when these conditions hold the limit is $a$.
