91Ó°ÊÓ

For the following exercises, find an equation of the level curve of \(f\) that contains the point \(P .\) $$ g(x, y)=y^{2} \arctan x, P(1,2) $$

For the following exercises, find an equation of the level curve of \(f\) that contains the point \(P .\) $$ g(x, y)=e^{x y}\left(x^{2}+y^{2}\right), P(1,0) $$

A thin plate made of iron is located in the \(x y\) -plane. The temperature \(T\) in degrees Celsius at a point \(P(x, y)\) is inversely proportional to the square of its distance from the origin. Express \(T\) as a function of \(x\) and \(y\)

For the following exercises, find the limit of the function. $$ \lim _{(x, y) \rightarrow(1,2)} x $$

For the following exercises, find the limit of the function. $$ \lim _{(x, y) \rightarrow(1,2) x^{2}+y^{2}} $$

For the following exercises, find the limit of the function. Show that the limit \(\lim _{(x, y) \rightarrow(0,0) x^{2}+y^{2}}\) exists and is the same along the paths: \(y\) -axis and \(x\) -axis, and along \(y=x\)

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit does not exist. $$ \lim _{(x, y) \rightarrow(0,0) 4 x^{2}-10 y^{2}+6} $$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit does not exist. $$ \lim _{(x, y) \rightarrow(11,13)} \sqrt{\frac{1}{x y}} $$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit does not exist. $$ \lim _{(x, y) \rightarrow(0,1)} \frac{y^{2} \sin x}{x} $$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit does not exist. $$ \lim _{(x, y) \rightarrow(0,0)} \sin \left(\frac{x^{8}+y^{7}}{x-y+10}\right) $$