91Ó°ÊÓ

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$f(x, y)=e^{3 x y}$$

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$f(x, y)=\frac{1}{x y}$$

For the following exercises, determine the region in which the function is continuous. Explain your answer. $$f(x, y)=\frac{x^{2} y}{x^{2}+y^{2}}$$

For the following exercises, determine the region in which the function is continuous. Explain your answer. $$f(x, y)=\left\\{\begin{array}{ll}\frac{x^{2} y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right\\}$$

For the following exercises, determine the region in which the function is continuous. Explain your answer. $$f(x, y)=\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$$

Determine whether \(g(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) is continuous at (0,0) .

Create a plot using graphing software to determine where the limit does not exist. Determine the region of the coordinate plane in which \(f(x, y)=\frac{1}{x^{2}-y}\) is continuous.

Determine the region of the \(x y\) -plane in which \(f(x, y)=\ln \left(x^{2}+y^{2}-1\right)\) is continuous. Use technology to support your conclusion. (Hint: Choose the range of values for \(x\) and \(y\) carefully!)

At what points in space is ist what \(g(x, y, z)=x^{2}+y^{2}-2 z^{2}\) continuous?

At what points in space is \(g(x, y, z)=\frac{1}{x^{2}+z^{2}-1}\) continuous?