[11.2,3] - Reading Assignment

After reading sections 11.2 and 11.3 ...

1. When talking about limits for functions of several variables, why isn't it sufficient to say $$\lim_{(x,y)\to(0,0)}f(x,y)=L$$ if $f(x,y)$ gets close to $L$ as we approach (0,0) along the $x$ axis and along the $y$ axis. (Hint: consider path independence.)
2. Show that $\lim_{(x,y)\to(0,0)}\(x^2+y^2\)=0$.
3. Consider $\lim_{(x,y)\to\infty}\( \frac{xy}{3x^2+2y^2}\)$. Why does the limit not exist?
4. Suppose the $f(x,y)$ is continuous everywhere. Assume that $f_x(1,1)=2$, $f_y(1,1)=-2$, and $f_{xy}(1,1)=3$. Can you compute $f_{yx}(1,1)$ from this information alone?
5. Muddy questions? Questions you wonder about?