This problem allows the student to see the actual formula for the function. Another version that does not give explicit formulas is available and probably is a better way to make the student consider concepts only.

Determining $$\lim_{(x,y)\rightarrow(x_0,y_0)} f(x,y)$$ by definition can often be very difficult. The interactive worksheet above allows the user to visualize this problem using a polar coordinates approach. Such an approach allows one to squeeze the domain values $(x,y)$ about the point of interest $(x_0,y_0)$. If the range values on the surface are then subsequently squeezed to a point, the function has a limit of this common value. If not, then no limit will exist.

Determine an appropriate value for each limit or (if no limit exists) enter NONE. (For these problems, presume any points not in the domain of the given function are not of interest.)

$$\lim_{(x,y)\rightarrow (0,0)} (2*(x^3-y^3)/(x^2+y^2)) =$$
$$\lim_{(x,y)\rightarrow (0,0)} ((x^2-3*y^2)/(x^2+y^2)) =$$
$$\lim_{(x,y)\rightarrow (1,0)} (8*(x^2-y^2)/(2*x^2+2*y^2)) =$$
$$\lim_{(x,y)\rightarrow (0,0)} (2*sin(x*y)/[(x+y)^2]) =$$
$$\lim_{(x,y)\rightarrow (0,0)} ([sin(4*(x^2+y^2+9*pi))]/(x^2+y^2)) =$$
$$\lim_{(x,y)\rightarrow (0,0)} ((x^2+y^2+9)/(x^2+y^2+2)) =$$

Hint: